In recent years, a wide range of mortality models has been proposed to address the diverse factors influencing mortality rates, which has highlighted the need to perform model selection. Traditional mortality model selection methods, such as AIC and BIC, often require fitting multiple models independently and ranking them based on these criteria. This process can fail to account for uncertainties in model selection, which can lead to overly optimistic prediction intervals, and it disregards the potential insights from combining models. To address these limitations, we propose a novel Bayesian model selection framework that integrates model selection and parameter estimation into the same process. This requires creating a model-building framework that will give rise to different models by choosing different parametric forms for each term. Inference is performed using the reversible jump Markov chain Monte Carlo algorithm, which is devised to allow for transition between models of different dimensions, as is the case for the models considered here. We develop modeling frameworks for data stratified by age and period and for data stratified by age, period, and product. Our results are presented in two case studies.

The matrixdist R package provides a comprehensive suite of tools for the statistical analysis of matrix distributions, including phase-type, inhomogeneous phase-type, discrete phase-type, and related multivariate distributions. This paper introduces the package and its key features, including the estimation of these distributions and their extensions through expectation-maximization algorithms, as well as the implementation of regression through the proportional intensities and mixture-of-experts models. Additionally, the paper provides an overview of the theoretical background, discusses the algorithms and methods implemented in the package, and offers practical examples to illustrate the application of matrixdist in real-world actuarial problems. The matrixdist R package aims to provide researchers and practitioners a wide set of tools for analyzing and modeling complex data using matrix distributions.

This paper examines an insurer’s optimal asset allocation and reinsurance policies. The financial market framework includes one risk-free and one risky asset. The insurer has two business lines, where the ordinary claim process is modeled by a compound Poisson process and catastrophic claims follow a compound dynamic contagion process. The dynamic contagion process, which is a generalization of the externally exciting Cox process with shot-noise intensity and the self-exciting Hawkes process, is enhanced by accommodating the dependency structure between the magnitude of contribution to intensity after initial events for catastrophic insurance products and its claim/loss size. We also consider the dependency structure between the positive effect on the intensity and the negative crashes on the risky financial asset when initial events occur. Our objective is to maximize the insurer’s expected utility of terminal surplus. We construct the extended Hamilton–Jacobi–Bellman (HJB) equation using dynamic programming principles to derive an explicit optimal reinsurance policy for ordinary claims. We further develop an iterative scheme for solving the value function and the optimal asset allocation policy and the reinsurance policy for catastrophic claims numerically, providing a rigorous convergence proof. Finally, we present numerical examples to demonstrate the impact of key parameters.

Accurate mortality forecasting is crucial for actuarial pricing, reserving, and capital planning, yet the traditional Lee-Carter model struggles with non-linear age and cohort patterns, coherent multi-population forecasting, and quantifying prediction uncertainties. Recent advances in deep learning provide a range of tools that can address these limitations, but actuarial surveys have not kept pace. This paper provides the first concise view of deep learning in mortality forecasting. We cover six deep network architectures, namely Recurrent Neural Networks, Convolutional Neural Networks, Transformers, Autoencoders, Locally Connected Networks, and Multi-Task Feed-Forward Networks. We discuss how these architectures tackle cohort effects, population coherence, interpretability, and uncertainty in mortality forecasting. Evidence from the literature shows that carefully calibrated deep learning models can consistently outperform the Lee-Carter baselines; however, no single architecture resolves every challenge, and open issues remain with data scarcity, interpretability, uncertainty quantification, and keeping pace with the advances of deep learning. This review is also intended to provide actuaries with a practical roadmap for adopting deep learning models in mortality forecasting.

Cyber breaches pose a significant threat to both enterprises and society. Analyzing cyber breach data is essential for improving cyber risk management and developing effective cyber insurance policies. However, modeling cyber risk is challenging due to its inherent characteristics, including sparsity, heterogeneity, heavy tails, and dependence. This work introduces a cluster-based dependence model that captures both temporal and cross-group dependencies, providing a more accurate representation of multivariate cyber breach risks. The proposed framework employs a cluster-based kernel approach to model breach severity, effectively handling heterogeneity and extreme values, while a copula-based method is used to capture multivariate dependence. Our findings, validated through both empirical and synthetic studies, demonstrate that the proposed model effectively captures the statistical characteristics of multivariate cyber breach risks and outperforms commonly used models in predictive accuracy. Furthermore, we show that our approach can enhance cyber insurance pricing by generating more profitable insurance contracts.

Disaster Risk Financing (DRF) presents a massive challenge to governments worldwide in protecting against catastrophic disaster losses. This study explores the development of a Disaster Fund that optimally integrates various DRF instruments, considering several real-world factors, including limited reserves, constrained risk horizons, risk aversion, risk tolerance, insurance structures, and premium pricing strategies. We demonstrate that the Value-at-Risk (VaR) and Tail VaR constraints are equivalent when the government has a limited risk horizon. Furthermore, we investigate the optimality of various insurance structures under different premium principles, conduct comparative statics on key parameters, and analyze the influence of a VaR constraint on the optimal mix of disaster financing instruments. Lastly, we apply our Disaster Fund model to the National Flood Insurance Program dataset to assess the optimal disaster financing strategy within the context of our framework.

Ponzi schemes are financial frauds that are pervasive throughout the world. Since they cause serious harm to society, it is of interest to study them so that they can be prevented. Typically, a Ponzi scheme is instigated by a promoter who promises above-average investment returns. He uses funds from the early investors to pay his later investors. These scams can occasionally last a long time, but they are ultimately unsustainable. This paper describes some well-known Ponzi schemes and identifies their common characteristics. We also review some of the approaches used to model Ponzi schemes.

Tree-based methods are widely used in insurance pricing due to their simple and accurate splitting rules. However, there is no guarantee that the resulting premiums avoid indirect discrimination when features recorded in the database are correlated with the protected variable under consideration. This paper shows that splitting rules in regression trees and random forests can be adapted in order to avoid indirect discrimination related to a binary protected variable like gender. The new procedure is illustrated on motor third-party liability insurance claim data.

The rise of large language models (LLMs) has marked a substantial leap toward artificial general intelligence. However, the utilization of LLMs in (re)insurance sector remains a challenging problem because of the gap between general capabilities and domain-specific requirements. Two prevalent methods for domain specialization of LLMs involve prompt engineering and fine-tuning. In this study, we aim to evaluate the efficacy of LLMs, enhanced with prompt engineering and fine-tuning techniques, on quantitative reasoning tasks within the (re)insurance domain. It is found that (1) compared to prompt engineering, fine-tuning with task-specific calculation dataset provides a remarkable leap in performance, even exceeding the performance of larger pre-trained LLMs; (2) when acquired task-specific calculation data are limited, supplementing LLMs with domain-specific knowledge dataset is an effective alternative; and (3) enhanced reasoning capabilities should be the primary focus for LLMs when tackling quantitative tasks, surpassing mere computational skills. Moreover, the fine-tuned models demonstrate a consistent aptitude for common-sense reasoning and factual knowledge, as evidenced by their performance on public benchmarks. Overall, this study demonstrates the potential of LLMs to be utilized as powerful tools to serve as AI assistants and solve quantitative reasoning tasks in (re)insurance sector.

We seek to understand the factors that drive mortality in the contiguous United States using data that are indexed by county and year and grouped into 18 different age bins. We propose a model that adds two important contributions to existing mortality studies. First, we treat age as a random effect. This is an improvement over previous models because it allows the model in one age group to borrow information from other age groups. Second, we utilize Gaussian Processes to create nonlinear covariate effects for predictors such as unemployment rate, race, and education level. This allows for a more flexible relationship to be modeled between mortality and these predictors. Understanding that the United States is expansive and diverse, we allow for many of these effects to vary by location. The flexibility in how predictors relate to mortality has not been used in previous mortality studies and will result in a more accurate model and a more complete understanding of the factors that drive mortality. Both the multivariate nature of the model as well as the spatially varying non-linear predictors will advance the study of mortality and will allow us to better examine the relationships between the predictors and mortality.

We investigate state-level age-specific mortality trends based on the United States Mortality Database (USMDB) published by the Human Mortality Database. In tandem with looking at the longevity experience across all the states, we also consider a collection of socio-demographic, economic, and educational covariates that correlate with mortality trends. To obtain smoothed mortality surfaces for each state, we implement the machine learning framework of Multi-Output Gaussian Process regression (Huynh & Ludkovski, AAS, 2021) on targeted groupings of 3–6 states. Our detailed exploratory analysis shows that the mortality experience is highly inhomogeneous across states in terms of respective Age structures. We moreover document multiple divergent trends between best and worst states, between Females and Males, and between younger and older Ages. The comparisons across the 50+ fitted models offer opportunities for rich insights about drivers of mortality in the U.S. and are visualized through numerous figures and an online interactive dashboard.

The classical credibility premium provides a simple and efficient method for predicting future damages and losses. However, when dealing with a nonhomogeneous population, this widely used technique has been challenged by the Regression Tree Credibility (RTC) model and the Logistic Regression Credibility (LRC) model. This article introduces the Mixture Credibility Formula (MCF), which represents a convex combination of the classical credibility premiums of several homogeneous subpopulations derived from the original population. We also compare the performance of the MCF method with the RTC and LRC methods. Our analysis demonstrates that the MCF method consistently outperforms these approaches in terms of the quadratic loss function, highlighting its effectiveness in refining insurance premium calculations and enhancing risk assessment strategies.

Healthcare costs tend to increase with age. In particular, in the case of illness, the last year before death can be an exceptionally costly period as the need for healthcare increases. Using a novel private insurance dataset containing over one million records of claims submitted by individuals to their health insurance providers during the last year of life, our research seeks to shed light on the costs before death in Switzerland. Our work documents how spending patterns change with proximity to dying. We use machine learning algorithms to identify and quantify the key effects that drive a person’s spending during this critical period. Our findings provide a more profound understanding of the costs associated with hospitalization before death, the role of age, and the variation in costs based on the services, including care services, which individuals require.

We consider a new approach in the definition of two-dimensional heavy-tailed distributions. Specifically, we introduce the classes of two-dimensional long-tailed, of two-dimensional dominatedly varying, and of two-dimensional consistently varying distributions. Next, we define the closure property with respect to two-dimensional convolution and to joint max-sum equivalence in order to study whether they are satisfied by these classes. Further, we examine the joint-tail behavior of two random sums, under generalized tail asymptotic independence. Afterward, we study the closure property under scalar product and two-dimensional product convolution, and by these results, we extended our main result in the case of jointly randomly weighted sums. Our results contained some applications where we establish the asymptotic expression of the ruin probability in a two-dimensional discrete-time risk model.

This paper utilizes neural networks (NNs) for cycle detection in the insurance industry. The efficacy of NNs is compared on simulated data to the standard methods used in the underwriting cycles literature. The results show that NN models perform well in detecting cycles even in the presence of outliers and structural breaks. The methodology is applied to a granular data set of prices per risk profile from the Brazilian insurance industry.

Many pension plans and private retirement products contain annuity factors, converting the funds at some future time into lifelong income. In general model settings like, for example, the Li-Lee mortality model, analytical values for the annuity factors are not available and one has to rely on numerical techniques. Their computation typically requires nested simulations as they depend on the interest rate level and the mortality tables at the time of retirement. We exploit the flexibility and efficiency of feed-forward neural networks (NNs) to value the annuity factors at the time of retirement. In a numerical study, we compare our deep learning approach to (least-squares) Monte-Carlo, which can be represented as a special case of the NN.

In this article, we study an optimization problem for a couple including two breadwinners with uncertain life times. Both breadwinners need to choose the optimal strategies for consumption, investment, housing, and life insurance purchasing to maximize the utility. In this article, the prices of housing assets and investment risky assets are assumed to be correlated. These two breadwinners are considered to have dependent mortality rates to include the breaking heart effect. The method of copula functions is used to construct the joint survival functions of two breadwinners. The analytical solutions of optimal strategies can be achieved, and numerical results are demonstrated.

We consider pricing of a specialised critical illness and life insurance contract for breast cancer (BC) risk. We compare (a) an industry-based Markov model with (b) a recently developed semi-Markov model, which accounts for unobserved BC cases and progression through clinical stages of BC, and (c) an alternative Markov model derived from (b). All models are calibrated using population data in England and data from the medical literature. We show that the semi-Markov model aligns best with empirical evidence. We then consider net premiums of specialized life insurance products under various scenarios of cancer diagnosis and treatment. The results show strong dependence on the time spent with diagnosed or undiagnosed pre-metastatic BC. This proves to be significant for refining cancer survival estimates and accurately estimating related age dependence by cancer stage. In contrast, the industry-based model, by overlooking this critical factor, is more sensitive to the model assumptions, underscoring its limitations in cancer estimates.

Cause-of-death mortality forecasting, a key topic in public health and actuarial science, is a challenging task due to the difficulty of modeling that accounts for dependencies among causes of death. While several cause-of-death mortality models have been proposed to address this difficulty, little attention has been paid to improving their predictive performance. Recently, purely data-driven approaches using tensor decomposition methods have been introduced to cause-of-death mortality modeling, demonstrating strong out-of-sample predictive performance compared to existing models. However, these methods have difficulties in the interpretability of multi-rank tensor components to achieve strong predictive performance. In response, we propose a novel tensor-based cause-of-death mortality model by replacing the tensor decomposition with a convolutional autoencoder with a one-dimensional latent layer that provides a Lee-Carter-like time-series factor; the model also provides the age sensitivity of cause-specific log mortality to the time-series factor. Due to the representational capability of the neural network, our model achieves better predictive performance compared to the existing tensor decomposition-based models, despite the simplified latent layer for model interpretability.

Understanding and forecasting mortality by cause is an essential branch of actuarial science, with wide-ranging implications for decision-makers in public policy and industry. To accurately capture trends in cause-specific mortality, it is critical to consider dependencies between causes of death and produce forecasts by age and cause coherent with aggregate mortality forecasts. One way to achieve these aims is to model cause-specific deaths using compositional data analysis (CODA), treating the density of deaths by age and cause as a set of dependent, nonnegative values that sum to one. A major drawback of standard CODA methods is the challenge of zero values, which frequently occur in cause-of-death mortality modeling. Thus, we propose using a compositional power transformation, the $\alpha$-transformation, to model cause-specific life-table death counts. The $\alpha$-transformation offers a statistically rigorous approach to handling zero value subgroups in CODA compared to ad hoc techniques: adding an arbitrarily small amount. We illustrate the $\alpha$-transformation in England and Wales and US death counts by cause from the Human Cause-of-Death database, for cardiovascular-related causes of death. The results demonstrate the $\alpha$-transformation improves forecast accuracy of cause-specific life-table death counts compared with log-ratio-based CODA transformations. The forecasts suggest declines in the proportions of deaths from major cardiovascular causes (myocardial infarction and other ischemic heart diseases).