The expanding application of advanced analytics in insurance has generated numerous opportunities, such as more accurate predictive modeling powered by machine learning and artificial intelligence (AI) methods, the utilization of novel and unstructured datasets, and the automation of key operations. Significant advances in these areas are being made through novel applications and adaptations of predictive modeling techniques for insurance purposes, while, concurrently, rapid advances in machine learning methods are being made outside of the insurance sector. However, these innovations also bring substantial challenges, particularly around the transparency, explanation, and fairness of complex algorithmic models and the economic and societal impacts of their adoption in decision-making. As insurance is a highly regulated industry, models may be required by regulators to be explainable, in order to enable analysis of the basis for decision making. Due to the societal importance of insurance, significant attention is being paid to ensuring that insurance models do not discriminate unfairly. In this special issue, we feature papers that explore key issues in insurance analytics, focusing on prediction, explainability, and fairness.

Handling nominal covariates with a large number of categories is challenging for both statistical and machine learning techniques. This problem is further exacerbated when the nominal variable has a hierarchical structure. We commonly rely on methods such as the random effects approach to incorporate these covariates in a predictive model. Nonetheless, in certain situations, even the random effects approach may encounter estimation problems. We propose the data-driven Partitioning Hierarchical Risk-factors Adaptive Top-down algorithm to reduce the hierarchically structured risk factor to its essence, by grouping similar categories at each level of the hierarchy. We work top-down and engineer several features to characterize the profile of the categories at a specific level in the hierarchy. In our workers’ compensation case study, we characterize the risk profile of an industry via its observed damage rates and claim frequencies. In addition, we use embeddings to encode the textual description of the economic activity of the insured company. These features are then used as input in a clustering algorithm to group similar categories. Our method substantially reduces the number of categories and results in a grouping that is generalizable to out-of-sample data. Moreover, we obtain a better differentiation between high-risk and low-risk companies.

In a subgroup analysis for an actuarial problem, the goal is for the investigator to classify the policyholders into unique groups, where the claims experience within each group are made as homogenous as possible. In this paper, we illustrate how the alternating direction method of multipliers (ADMM) approach for subgroup analysis can be modified so that it can be more easily incorporated into an insurance claims analysis. We present an approach to penalize adjacent coefficients only and show how the algorithm can be implemented for fast estimation of the parameters. We present three different cases of the model, depending on the level of dependence among the different coverage groups within the data. In addition, we provide an interpretation of the credibility problem using both random effects and fixed effects, where the fixed effects approach corresponds to the ADMM approach to subgroup analysis, while the random effects approach represents the classic Bayesian approach. In an empirical study, we demonstrate how these approaches can be applied to real data using the Wisconsin Local Government Property Insurance Fund data. Our results show that the presented approach to subgroup analysis could provide a classification of the policyholders that improves the prediction accuracy of the claim frequencies in case other classifying variables are unavailable in the data.

Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. Wüthrich and Buser (Data Analytics for Non-Life Insurance Pricing. Lecture notes available at SSRN. http://dx.doi.org/10.2139/ssrn.2870308, 2019) established that boosting can be conducted directly on the response under Poisson deviance loss function and log-link, by adapting the weights at each step. This is particularly useful to analyze low counts (typically, numbers of reported claims at policy level in personal lines). Huyghe et al. (Boosting cost-complexity pruned trees on Tweedie responses: The ABT machine for insurance ratemaking. Scandinavian Actuarial Journal. https://doi.org/10.1080/03461238.2023.2258135, 2022) adopted this approach to propose a new boosting machine with cost-complexity pruned trees. In this approach, trees included in the score progressively reduce to the root-node one, in an adaptive way. This paper reviews these results and presents the new BT package in R contributed by Willame (Boosting Trees Algorithm. https://cran.r-project.org/package=BT; https://github.com/GiregWillame/BT, 2022), which is designed to implement this approach for insurance studies. A numerical illustration demonstrates the relevance of the new tool for insurance pricing.

As the global population continues to age, effective management of longevity risk becomes increasingly critical for various stakeholders. Accurate mortality forecasting serves as a cornerstone for addressing this challenge. This study proposes to leverage Kernel Principal Component Analysis (KPCA) to enhance mortality rate predictions. By extending the traditional Lee-Carter model with KPCA, we capture nonlinear patterns and complex relationships in mortality data. The newly proposed KPCA Lee-Carter algorithm is empirically tested and demonstrates superior forecasting performance. Furthermore, the model’s robustness was tested during the COVID-19 pandemic, showing that the KPCA Lee-Carter algorithm effectively captures increased uncertainty during extreme events while maintaining narrower prediction intervals. This makes it a valuable tool for mortality forecasting and risk management. Our findings contribute to the growing body of literature where actuarial science intersects with statistical learning, offering practical solutions to the challenges posed by an aging world population.

In this paper, we construct interpretable zero-inflated neural network models for modeling hospital admission counts related to respiratory diseases among a health-insured population and their dependants in the United States. In particular, we exemplify our approach by considering the zero-inflated Poisson neural network (ZIPNN), and we follow the combined actuarial neural network (CANN) approach for developing zero-inflated combined actuarial neural network (ZIPCANN) models for modeling admission rates, which can accommodate the excess zero nature of admission counts data. Furthermore, we adopt the LocalGLMnet approach (Richman & Wüthrich (2023). Scandinavian Actuarial Journal, 2023(1), 71–95.) for interpreting the ZIPNN model results. This facilitates the analysis of the impact of a number of socio-demographic factors on the admission rates related to respiratory disease while benefiting from an improved predictive performance. The real-life utility of the methodologies developed as part of this work lies in the fact that they facilitate accurate rate setting, in addition to offering the potential to inform health interventions.

The paper introduces a method for creating a categorical generalized linear model (GLM) based on information extracted from a given black-box predictor. The procedure for creating the guided GLM is as follows: For each covariate, including interactions, a covariate partition is created using partial dependence functions calculated based on the given black-box predictor. In order to enhance the predictive performance, an auto-calibration step is used to determine which parts of each covariate partition should be kept, and which parts should be merged. Given the covariate and interaction partitions, a standard categorical GLM is fitted using a lasso penalty. The performance of the proposed method is illustrated using a number of real insurance data sets where gradient boosting machine (GBM) models are used as black-box reference models. From these examples, it is seen that the predictive performance of the guided GLMs is very close to that of the corresponding reference GBMs. Further, in the examples, the guided GLMs have few parameters, making the resulting models easy to interpret. In the numerical illustrations techniques are used to, e.g., identify important interactions both locally and globally, which is essential when, e.g., constructing a tariff.

Generalized additive models (GAMs) are a leading model class for interpretable machine learning. GAMs were originally defined with smooth shape functions of the predictor variables and trained using smoothing splines. Recently, tree-based GAMs where shape functions are gradient-boosted ensembles of bagged trees were proposed, leaving the door open for the estimation of a broader class of shape functions (e.g. Explainable Boosting Machine (EBM)). In this paper, we introduce a competing three-step GAM learning approach where we combine (i) the knowledge of the way to split the covariates space brought by an additive tree model (ATM), (ii) an ensemble of predictive linear scores derived from generalized linear models (GLMs) using a binning strategy based on the ATM, and (iii) a final GLM to have a prediction model that ensures auto-calibration. Numerical experiments illustrate the competitive performances of our approach on several datasets compared to GAM with splines, EBM, or GLM with binarsity penalization. A case study in trade credit insurance is also provided.

Deep neural networks have become an important tool for use in actuarial tasks, due to the significant gains in accuracy provided by these techniques compared to traditional methods, but also due to the close connection of these models to the generalized linear models (GLMs) currently used in industry. Although constraining GLM parameters relating to insurance risk factors to be smooth or exhibit monotonicity is trivial, methods to incorporate such constraints into deep neural networks have not yet been developed. This is a barrier for the adoption of neural networks in insurance practice since actuaries often impose these constraints for commercial or statistical reasons. In this work, we present a novel method for enforcing constraints within deep neural network models, and we show how these models can be trained. Moreover, we provide example applications using real-world datasets. We call our proposed method ICEnet to emphasize the close link of our proposal to the individual conditional expectation model interpretability technique.