In machine learning-based mortality models, interpretation methods are well established, and they can reveal structures resembling the age or time effects in traditional mortality models. However, in the reverse direction, using such traditional components to guide the initialization of a neural network remains highly challenging due to information loss during model interpretation. This study addresses this gap by exploring how components from pre-fitted traditional mortality models can be used to initialize neural networks, enabling structural information to be incorporated into a deep learning framework. We introduce Kolmogorov–Arnold Networks (KAN) and first construct two shallow models, KAN[2,1] and ARIMAKAN, to examine their applicability to mortality modeling. We then extend the Combined Actuarial Neural Network (CANN) into a KAN-based Actuarial Neural Network (KANN), in which classical model components calibrated via generalized nonlinear models or generalized additive models are naturally used for initialization. Three KANN variants, namely KANN[2,1], KANNLC, and KANNAPC, are proposed. In these models, neural networks assist in improving the accuracy of traditional models and help refine the original parameter estimates. All KANN-based models can also produce smooth mortality curves as well as smooth age, period, and cohort effects through simple regularization. Experiments on 34 populations demonstrate that KAN-based approaches achieve stable performance while balancing interpretability, smoothness, and predictive accuracy.

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.

This article introduces a blockchain-based insurance scheme that integrates parametric and collaborative elements. A pool of investors, referred to as surplus providers, locks funds in a smart contract, enabling blockchain users to underwrite parametric insurance contracts. These contracts automatically trigger compensation when predefined conditions are met. The collaborative aspect is embodied in the generation of tokens, which are distributed to surplus providers. These tokens represent each participant’s share of the surplus and grant voting rights for management decisions. The smart contract is developed in Solidity, a high-level programming language for the Ethereum blockchain, and deployed on the Sepolia testnet, with data processing and analysis conducted using Python. In addition, open-source code is provided and main research challenges are identified, so that further research can be carried out to overcome limitations of this first proof of concept.

The limited stop-loss transform, along with the stop-loss and limited loss transforms – which are special or limiting cases of the limited stop-loss transform – is one of the most important transforms used in insurance, and it also appears extensively in many other fields including finance, economics, and operations research. When the distribution of the underlying loss is uncertain, the worst-case risk measure for the limited stop-loss transform plays a key role in many quantitative risk management problems in insurance and finance. In this paper, we derive expressions for the worst-case distortion risk measure of the limited stop-loss transform, as well as for the stop-loss and limited loss transforms, when the distribution of the underlying loss is uncertain and lies in a general $k$-order Wasserstein ball that contains a reference distribution. We also identify the worst-case distributions under which the worst-case distortion risk measures are attained. Additionally, our results also recover the findings of Guan et al. ((2023) North American Actuarial Journal, 28(3), 611–625), regarding the worst-case stop-loss premium over a $k$-order Wasserstein ball. Furthermore, we use numerical examples to illustrate the worst-case distributions and the worst-case risk measures derived in this paper. We also examine the effects of the reference distribution, the radius of the Wasserstein ball, and the retention levels of limited stop-loss reinsurance on the premium for this type of reinsurance.

In this paper, we investigate asymmetric Nash bargaining in the context of proportional insurance contracts between a risk-averse insured and a risk-averse insurer, both seeking to enhance their expected utilities. We obtain a necessary and sufficient condition for the Pareto optimality of the status quo and derive the optimal Nash bargaining solution when the status quo is Pareto dominated. If the insured’s and the insurer’s risk preference exhibit decreasing absolute risk aversion and the insurer’s initial wealth decreases in the insurable risk in the sense of reversed hazard rate order, we show that both the optimal insurance coverage and the optimal insurance premium increase with the insured’s degree of risk aversion and the insurer’s bargaining power. If the insured’s risk preference further follows constant absolute risk aversion, we find that greater insurance coverage is induced as the insurer’s constant initial wealth increases.

In this paper, we study a two-period optimal insurance problem for a policyholder with mean-variance preferences who purchases proportional insurance at the beginning of each period. The insurance premium is calculated by a variance premium principle with a risk loading that depends on the policyholder’s claim history. We derive the time-consistent optimal insurance strategy in closed form and the optimal constant precommitment strategy in semiclosed form. For the optimal general precommitment strategy, we obtain the solution for the second period semi-explicitly and, then, the solution for the first period numerically via an efficient algorithm. Furthermore, we compare the three types of optimal strategies, highlighting their differences, and we examine the impact of the key model parameters on the optimal strategies and value functions.

We introduce a novel class of bivariate common-shock discrete phase-type (CDPH) distributions to describe dependencies in loss modeling, with an emphasis on those induced by common shocks. By constructing two jointly evolving terminating Markov chains that share a common evolution up to a random time corresponding to the common shock component, and then proceed independently, we capture the essential features of risk events influenced by shared and individual-specific factors. We derive explicit expressions for the joint distribution of the termination times and prove various class and distributional properties, facilitating tractable analysis of the risks. Extending this framework, we model random sums where aggregate claims are sums of continuous phase-type random variables with counts determined by these termination times and show that their joint distribution belongs to the multivariate phase-type or matrix-exponential class. We develop estimation procedures for the CDPH distributions using the expectation-maximization algorithm and demonstrate the applicability of our models through simulation studies and an application to bivariate insurance claim frequency data. In particular, the distribution of the latent common shock component present in correlated count data can be estimated as well.

The bonus-malus system (BMS) is a widely recognized and commonly employed risk management tool. A well-designed BMS can match expected insurance payments with estimated claims even in a diverse group of risks. Although there has been abundant research on improving bonus-malus (BM) systems, one important aspect has been overlooked: the stationary probability of a BMS satisfies the monotone likelihood ratio property. The monotone likelihood ratio for stationary probabilities allows us to better understand how riskier policyholders are more likely to remain in higher premium categories, while less risky policyholders are more likely to move toward lower premiums. This study establishes this property for BMSs that are described by an ergodic Markov chain with one possible claim and a transition rule +1/-d. We derive this result from the linear recurrences that characterize the stationary distribution; this represents a novel analytical approach in this domain. We also illustrate the practical implications of our findings: in the BM design problem, the premium scale is automatically monotonic.

Introduced over a century ago, Whittaker–Henderson smoothing remains widely used by actuaries in constructing one-dimensional and two-dimensional experience tables for mortality, disability, and other life insurance risks. In this paper, we reinterpret this smoothing technique within a modern statistical framework and address six practically relevant questions about its use. First, we adopt a Bayesian perspective on this method to construct credible intervals. Second, in the context of survival analysis, we clarify how to choose the observation and weight vectors by linking the smoothing technique to a maximum likelihood estimator. Third, we improve accuracy by relaxing the method’s reliance on an implicit normal approximation. Fourth, we select the smoothing parameters by maximizing a marginal likelihood function. Fifth, we improve computational efficiency when dealing with numerous observation points and consequently parameters. Finally, we develop an extrapolation procedure that ensures consistency between estimated and predicted values through constraints.

We propose a novel micro-level Cox model for incurred but not reported (IBNR) claims count based on hidden Markov models. Initially formulated as a continuous-time model, it addresses the complexity of incorporating temporal dependencies and policyholder risk attributes. However, the continuous-time model faces significant challenges in maximizing the likelihood and fitting right-truncated reporting delays. To overcome these issues, we introduce two discrete-time versions: one incorporating unsystematic randomness in reporting delays through a Dirichlet distribution and one without. We provide the EM algorithm for parameter estimation for all three models and apply them to an auto-insurance dataset to estimate IBNR claim counts. Our results show that while all models perform well, the discrete-time versions demonstrate superior performance by jointly modeling delay and frequency, with the Dirichlet-based model capturing additional variability in reporting delays. This approach enhances the accuracy and reliability of IBNR reserving, offering a flexible framework adaptable to different levels of granularity within an insurance portfolio.

We introduce a new class of heavy-tailed distributions for which any weighted average of independent and identically distributed random variables is larger than one such random variable in (usual) stochastic order. We show that many commonly used extremely heavy-tailed (i.e., infinite-mean) distributions, such as the Pareto, Fréchet, and Burr distributions, belong to this class. The established stochastic dominance relation can be further generalized to allow negatively dependent or non-identically distributed random variables. In particular, the weighted average of non-identically distributed random variables dominates their distribution mixtures in stochastic order.