Gaussian Process (GP) modeling is a probabilistic, non-parametric framework for describing spatio-temporal dependence that is well-suited for fitting risk-related surfaces. I summarize the main emerging actuarial use cases of GPs, including their applications in longevity modeling, insurance contract valuation, and loss development. The editorial also discusses further contexts with potential for GP-based approaches.

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.

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.

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.

Fine-grained mortality forecasting has gained momentum in actuarial research due to its ability to capture localized, short-term fluctuations in death rates. This paper introduces MortFCNet, a deep-learning method that predicts weekly death rates using region-specific weather inputs. Unlike traditional Serfling-based methods and gradient-boosting models that rely on predefined fixed Fourier terms and manual feature engineering, MortFCNet automatically learns patterns from raw time-series data without needing explicitly defined Fourier terms or manual feature engineering. Extensive experiments across over 200 NUTS-3 regions in France, Italy, and Switzerland demonstrate that MortFCNet consistently outperforms both a standard Serfling-type baseline and XGBoost in terms of predictive accuracy. Our ablation studies further confirm its ability to uncover complex relationships in the data without feature engineering. Moreover, this work underscores a new perspective on exploring deep learning for advancing fine-grained mortality forecasting.

When overdispersion and correlation co-occur in longitudinal count data, as is often the case, an analysis method that can handle both phenomena simultaneously is needed. The correlated Poisson distribution (CPD) proposed by Drezner and Farnum (Communications in Statistics-Theory and Methods, 22(11), 3051–3063, 1994) is a generalization of the classical Poisson distribution with the incorporation of an additional parameter that allows dependence between successive observations of the phenomenon under study. This parameter both measures the correlation and reflects the degree of dispersion. The classical Poisson distribution is obtained as a special case when the correlation is zero. We present an in-depth review of this CPD and discuss some methods to estimate the distribution parameters. The inclusion of regression components in this distribution is enabled by allowing one of the parameters to include available information concerning, in this case, automobile insurance policyholders. The proposed distribution can be viewed as an alternative to the Poisson, negative binomial, and Poisson-inverse Gaussian approaches. We then describe applications of the distribution, suggest it is appropriate for modeling the number of claims in an automobile insurance portfolio, and establish some new distribution properties.

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.

Credibility theory provides a fundamental framework in actuarial science for estimating policyholder premiums by blending individual claims experience with overall portfolio data. Bühlmann and Bühlmann–Straub credibility models are widely used because, in the Bayesian hierarchical setting, they are the best linear Bayes estimators, minimizing the Bayes risk (expected squared error loss) within the class of linear estimators given the experience data for a particular risk class. To improve estimation accuracy, quadratic credibility models incorporate higher-order terms, capturing more information about the underlying risk structure. This study develops a robust quadratic credibility (RQC) framework that integrates second-order polynomial adjustments of robustly transformed ground-up loss data, such as winsorized moments, to improve stability in the presence of extreme claims or heavy-tailed distributions. Extending semi-linear credibility, RQC maintains interpretability while enhancing statistical efficiency. We establish its asymptotic properties, derive closed-form expressions for the RQC premium, and demonstrate its superior performance in reducing mean square error (MSE). We additionally derive semi-linear credibility structural parameters using winsorized data, further strengthening the robustness of credibility estimation. Analytical comparisons and empirical applications highlight RQC’s ability to capture claim heterogeneity, offering a more reliable and equitable approach to premium estimation. This research advances credibility theory by introducing a refined methodology that balances efficiency, robustness, and practical applicability across diverse insurance settings.

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 employ an appropriate change of measure technique to offer a general result connecting a general form of the Gerber–Shiu function with the distribution of the deficit at ruin under the new (exponentially tilted) measure. Exploiting this result, we extract closed-form formulae for special forms of the Gerber–Shiu function assuming two cases of bivariate distributions that describe the dependence structure between claim sizes and inter-claim times. More specifically, initially, we employ the Downton–Moran bivariate exponential distribution, and we offer explicit formulae for cases of the Gerber–Shiu functions that include the time and the number of claims until ruin. In addition, we derive a closed formula for the defective discounted joint density of the number of claims until ruin, the deficit at ruin, and the time until ruin. The same is achieved for the joint density of the number of claims and the deficit at ruin. We further generalize these results by assuming that the inter-claim times and the claim sizes follow a Kibble–Moran bivariate Erlang distribution. Finally, we offer numerical examples in order to illustrate our main results.

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.

We revisit the question of how to include parameter uncertainty in univariate parametric models of losses and loss ratios. We first review the statistical theory for including parameter uncertainty based on right Haar priors (RHPs), which applies to many commonly used models. In this theory, the prior is chosen in such a way as to ensure matching between predicted probabilities and the relative frequencies of future outcomes in repeated tests. This property is known as reliability, or calibration. We then test priors for including parameter uncertainty in a number of models not covered by RHP theory. For these models, we find priors that generate predictions that are more reliable than predictions based on maximum likelihood, although they are not perfectly reliable. We discuss numerical schemes that can be used to generate Bayesian predictions, including a novel use of asymptotic expansions, and we include an example in which we show the impact of including parameter uncertainty in the modeling of extreme hurricane losses. The tail loss estimates show material increases due to the inclusion of parameter uncertainty. Finally, we describe a new software library that makes it straightforward to apply the methods we describe.

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.

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.

We evaluate the performance and level of intergenerational cross-subsidy in flat-accrual and dynamic-accrual collective defined contribution (CDC) schemes, which have been designed to be compatible with UK legislation. In the flat-accrual scheme, all members accrue the benefits at the same rate, irrespective of age. This captures the most significant feature of the Royal Mail Collective Pension Plan, which is currently the only UK CDC scheme. The dynamic-accrual schemes seek to reduce intergenerational cross-subsidies by varying the rate of benefit-accrual schemes in accordance with the age of members and the current funding level. We find that these CDC schemes can often be successful in smoothing pension outcomes postretirement while outperforming a defined contribution scheme followed by annuity purchase at the point of retirement. However, this out-performance is not guaranteed in a flat-accrual scheme, and there is little smoothing of projected pension outcomes before retirement. There are significant intergenerational cross-subsidies in the flat-accrual scheme, which qualitatively mirror the cross-subsidies seen in defined benefit schemes, but the magnitude of cross-subsidies is much larger in flat-accrual CDC schemes. The dynamic-accrual scheme design seeks to reduce such cross-subsidies, but we find significant cross-subsidies still arise due to the approximate pricing methodology used to determine the benefits.

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.

The InsurTech industry has undergone almost a decade of development. Despite its initial success, the industry now faces challenges from global uncertainties and regulatory adjustments, which lead to concerns about sustainable profit growth and the ongoing development of InsurTech. This study provides an overview of the evolution of InsurTech development from both academic and practical perspectives. A bibliometric analysis of more than 20,000 published articles, including both practice articles and academic articles, is put forward. As compared to other review articles in this field, which often focus on either the practice or the scholarly side of development, this article brings together a review of both academic and practice-based articles from fields relevant to InsurTech including artificial intelligence, the Internet of Things, and also powerful computing technology. A keyword extraction framework is developed and applied. Using text analysis, this study reviews the prioritized topics, analyzes the robustness of the development of publication growth, identifies emerging insurance business lines, and also highlights the challenges and gaps in both academic and practice development. This study aims to motivate collaboration between academics and industry to face the challenges posed by the integration of InsurTech into insurance operations.

The practice of actuarial science has always been rooted in computation. From the early days of hand-constructed tables and commutation functions to today’s large-scale stochastic simulations and machine learning models, actuaries have continuously adapted their analytical tools to the technology of their time. The rapid growth of high-performance computing, open-source software, and data-driven methodologies now offers new possibilities for actuarial modeling – transforming not only how we calculate, but also how we think about risk, uncertainty, and decision-making. This editorial introduces a thematic collection on Actuarial Software, which showcases recent advances at the intersection of actuarial modeling and computational science.

The Nelson–Siegel model is widely used in fixed income markets to produce yield curve dynamics. The multiple time-dependent parameter model conveniently addresses the level, slope, and curvature dynamics of the yield curves. In this study, we present a novel state-space functional regression model that incorporates a dynamic Nelson–Siegel (DNS) model and functional regression formulations applied to a multi-economy setting. This framework offers distinct advantages in explaining the relative spreads in yields between a reference economy and a response economy. To address the inherent challenges of model calibration, a kernel principal component analysis is employed to transform the representation of functional regression into a finite-dimensional, tractable estimation problem. A comprehensive empirical analysis is conducted to assess the efficacy of the functional regression approach, including an in-sample performance comparison with the DNS model. We conducted the stress testing analysis of the yield curves’ term structure within a dual economy framework. The bond ladder portfolio was examined through a case study focused on spread modeling using historical data for US Treasury and UK bonds.