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.

A common narrative among insurance actuaries and business economists is that national or regional pension systems can be finetuned, optimized, and improved simply by tinkering with demographic and financial parameters; all within the context of the “right” mathematical model. Indeed, recent papers in the actuarial literature have offered technical fixes around savings rates, retirement ages, decumulation strategies as well as more refined mortality and interest rate models. But alas, not everything in the world of pensions and retirement can be optimized, in particular as it relates to the history, background culture, or religion of the underlying population.

This paper documents a statistically significant relationship between a region’s pension plan “health status” and the fraction of the region’s population identifying as Protestant Christians (PC). We begin the analysis at the national level using a well-known pension quality index and then obtain similar results for the actuarial funded status of U.S. state pension plans.

Overall, this work is within the sphere of recent literature that indicates historical religious beliefs, values, and culture matter for financial economic outcomes; a factor which obviously can’t be optimized within a mathematical Hamilton–Jacobi–Bellman (HJB) equation. In other words, some things in retirement are truly beyond control.

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.

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.

We study a discrete-time life cycle retirement planning problem for individual workers with four distinct investment options: self-management with dynamic investment (S), self-management with benchmark investment (B), hire-management with flexible allocation ($\text{H}_{1}$), and hire-management with alpha focus ($\text{H}_{2}$). We examine the investment strategies and consumption patterns during the defined contribution fund accumulation period, ending with a life annuity purchase at retirement to finance post-retirement consumption. Based on the calibrated model using US data, we employ numerical dynamic programming technique to optimize worker’s financial decisions. Our analysis reveals that, despite the agency risk, delegated investments can add value to a worker’s lifetime utility, with the $\text{H}_2$ option yielding the best lifetime utility outcome. However, after taking the fund management fee into consideration, we find that both the $\text{H}_1$ and $\text{H}_2$ options may not offer additional value compared to the S option, yet they still surpass the B option in performance.

We design the insurance contract when the insurer faces arson-type risks. We show that the optimal contract must be manipulation-proof. Therefore, it is continuous, has a bounded slope, and satisfies the no-sabotage condition when arson-type actions are free. Any contract that mixes a deductible, coinsurance, and an upper limit is manipulation-proof. A key feature of our models is that we provide a simple, general, and entirely elementary proof of manipulation-proofness that is easily adapted to different settings. We also show that the ability to perform arson-type actions reduces the insured’s welfare as less coverage is offered in equilibrium.

In this paper, we provide a theoretical framework justifying the existence of a correlation risk premium in a market with two traded assets. We prove that risk-neutral dependence can differ substantially from real-world dependence by characterizing the set of risk-neutral martingale measures. This implies that implied correlation can be significantly different with the realized correlation. Depending on the choice of the market regarding the pricing measure, implied correlation can be high or low. We label the difference between risk-neutral and real-world correlation the “correlation gap” and make the connection with correlation risk premium. We show how dispersion trading can be used to exploit this correlation gap and demonstrate how there can exist a negative correlation risk premium in the financial market.

The extended $3/2$ short rate model is a mean reverting model of the short rate which, for suitably chosen parameters, permits a sensible term structure of bond yields and closed-form valuation formulae of zero-coupon bonds and options. This article supplies proofs of the formulae for the expected present values of future cash flows under the real-world probability measure, known as actuarial valuation. Finally, we give formulae for asymptotic levels of bond yields and formulae for bond option prices for the extended $3/2$ model, under particular conditions on its parameters.

Aggregate implements an efficient fast Fourier transform (FFT)-based algorithm to approximate compound probability distributions. Leveraging FFT-based methods offers advantages over recursion and simulation-based approaches, providing speed and accuracy to otherwise time-consuming calculations. Combining user-friendly features and an expressive domain-specific language called DecL, Aggregate enables practitioners and nonprogrammers to work with complex distributions effortlessly. The software verifies the accuracy of its FFT-based numerical approximations by comparing their first three moments to those calculated analytically from the specified frequency and severity. This moment-based validation, combined with carefully chosen default parameters, allows users without in-depth knowledge of the underlying algorithm to be confident in the results. Aggregate supports a wide range of frequency and severity distributions, policy limits and deductibles, and reinsurance structures and has applications in pricing, reserving, risk management, teaching, and research. It is written in Python.

We investigate bottom-up risk aggregation applied by insurance companies facing reserve risk from multiple lines of business. Since risk capitals should be calculated in different time horizons and calendar years, depending on the regulatory or reporting regime (Solvency II vs IFRS 17), we study correlations of ultimate losses and correlations of one-year losses in future calendar years in lines of business. We consider a multivariate version of a Hertig’s lognormal model and we derive analytical formulas for the ultimate correlation and the one-year correlations in future calendar years. Our main conclusion is that the correlation coefficients that should be used in a bottom-up aggregation formula depend on the time horizon and the future calendar year where the risk emerges. We investigate analytically and numerically properties of the ultimate and the one-year correlations, their possible values observed in practice, and the impact of misspecified correlations on the diversified risk capital.

Generalized Poisson (GP) distribution was introduced in Consul & Jain ((1973). Technometrics, 15(4), 791–799.). Since then it has found various applications in actuarial science and other areas. In this paper, we focus on the distributional properties of GP and its related distributions. In particular, we study the distributional properties of distributions in the $\mathcal{H}$ family, which includes GP and generalized negative binomial distributions as special cases. We demonstrate that the moment and size-biased transformations of distributions within the $\mathcal{H}$ family remain in the same family, which significantly extends the results presented in Ambagaspitiya & Balakrishnan ((1994). ASTINBulletin: the Journal of the IAA, 24(2), 255–263.) and Ambagaspitiya ((1995). Insurance Mathematics and Economics, 2(16), 107–127.). Such findings enable us to provide recursive formulas for evaluating risk measures, such as Value-at-Risk and conditional tail expectation of the compound GP distributions. In addition, we show that the risk measures can be calculated by making use of transform methods, such as fast Fourier transform. In fact, the transformation method showed a remarkable time advantage over the recursive method. We numerically compare the risk measures of the compound sums when the primary distributions are Poisson and GP. The results illustrate the model risk for the loss frequency distribution.

Risk measurement and econometrics are the two pillars of actuarial science. Unlike econometrics, risk measurement allows taking into account decision-makers’ risk aversion when analyzing the risks. We propose a hybrid model that captures decision-makers’ regression-based approach to study risks, focusing on explanatory variables while paying attention to risk severity. Our model considers different loss functions that quantify the severity of the losses that are provided by the risk manager or the actuary. We present an explicit formula for the regression estimators for the proposed risk-based regression problem and study the proposed results. Finally, we provide a numerical study of the results using data from the insurance industry.

This paper presents the AffineMortality R package which performs parameter estimation, goodness-of-fit analysis, simulation, and projection of future mortality rates for a set of affine mortality models for use in pricing and reserving. The computational routines build on the univariate Kalman Filtering approach of Koopman and Durbin ((2000). Journal of Time Series Analysis, 21(3), 281–296.) along other numerical methods to enhance the robustness of the results. This paper provides a discussion of how the package works in order to effectively estimate and project survival curves, and describes the available functions. Illustration of the package for mortality analysis of the US male data set is provided.

This study investigates the benefits and drawbacks of pension plan consolidation by quantifying the impact of mergers of heterogeneous plans on different stakeholders in a unique Canadian implementation of defined benefit plans. Using a comprehensive framework that combines a realistic economic scenario generator, a stochastic mortality model that captures differences among subpopulations, a cost model with economies of scale, and a dynamic asset allocation methodology, we evaluate the combined effect of asset- and liability-side changes on three groups of measures: plan-related risk measures assessing profits from an economic capital perspective, consumption-based metrics to understand the impact on members, and contribution risk measures capturing the risk from the employer’s viewpoint. We apply the framework to a hypothetical and empirically relevant merger and find that consolidation is favorable under most circumstances: the positive impacts of better diversification and economies of scale continue to outweigh the negative effects of heterogeneity even when the merging plans have different mortality expectations, different maturity levels, or modest differences in initial funded ratios.

Based on the recent papers, two distributions for the total claims amount (loss cost) are considered: compound Poisson-gamma and Tweedie. Each is used as an underlying distribution in the Bonus-Malus Scale (BMS) model. The BMS model links the premium of an insurance contract to a function of the insurance experience of the related policy. In other words, the idea is to model the increase and the decrease in premiums for insureds who do or do not file claims. We applied our approach to a sample of data from a major insurance company in Canada. Data fit and predictability were analyzed. We showed that the studied models are exciting alternatives to consider from a practical point of view, and that predictive ratemaking models can address some important practical considerations.

This paper studies a hierarchical risk model where an accident can cause a combination of different types of claims, whose sizes could be dependent. In addition, the frequencies of accidents that cause the different combinations of claims are dependent. We first derive formulas for computing risk measures, such as the Tail Conditional Expectation and Tail Variance of the aggregate losses for a portfolio of businesses. Then, we present formulas for performing the associated capital allocation to different types of claims in the portfolio. The main tool we used is the moment (or size-biased) transform of the multivariate distributions.

This paper will outline the functionality available in the CovRegpy package which was written for actuarial practitioners, wealth managers, fund managers, and portfolio analysts in the language of Python 3.11. The objective is to develop a new class of covariance regression factor models for covariance forecasting, along with a library of portfolio allocation tools that integrate with this new covariance forecasting framework. The novelty is in two stages: the type of covariance regression model and factor extractions used to construct the covariates used in the covariance regression, along with a powerful portfolio allocation framework for dynamic multi-period asset investment management.

The major contributions of package CovRegpy can be found on the GitHub repository for this library in the scripts: CovRegpy.py, CovRegpy_DCC.py, CovRegpy_RPP.py, CovRegpy_SSA.py, CovRegpy_SSD.py, and CovRegpy_X11.py. These six scripts contain implementations of software features including multivariate covariance time series models based on the regularized covariance regression (RCR) framework, dynamic conditional correlation (DCC) framework, risk premia parity (RPP) weighting functions, singular spectrum analysis (SSA), singular spectrum decomposition (SSD), and X11 decomposition framework, respectively.

These techniques can be used sequentially or independently with other techniques to extract implicit factors to use them as covariates in the RCR framework to forecast covariance and correlation structures and finally apply portfolio weighting strategies based on the portfolio risk measures based on forecasted covariance assumptions. Explicit financial factors can be used in the covariance regression framework, implicit factors can be used in the traditional explicit market factor setting, and RPP techniques with long/short equity weighting strategies can be used in traditional covariance assumption frameworks.

We examine, herein, two real-world case studies for actuarial practitioners. The first of these is a modification (demonstrating the regularization of covariance regression) of the original example from Hoff & Niu ((2012). Statistica Sinica, 22(2), 729–753) which modeled the covariance and correlative relationship that exists between forced expiratory volume (FEV) and age and FEV and height. We examine this within the context of making probabilistic predictions about mortality rates in patients with chronic obstructive pulmonary disease.

The second case study is a more complete example using this package wherein we present a funded and unfunded UK pension example. The decomposition algorithm isolates high-, mid-, and low-frequency structures from FTSE 100 constituents over 20 years. These are used to forecast the forthcoming quarter’s covariance structure to weight the portfolio based on the RPP strategy. These fully funded pensions are compared against the performance of a fully unfunded pension using the FTSE 100 index performance as a proxy.

This paper introduces DivFolio, a multiperiod portfolio selection and analytic software application that incorporates automated and user-determined divestment practices accommodating Environmental Social Governance (ESG) and portfolio carbon footprint considerations. This freely available portfolio analytics software tool is written in R with a GUI interface developed as an R Shiny application for ease of user experience. Users can utilize this software to dynamically assess the performance of asset selections from global equity, exchange-traded funds, exchange-traded notes, and depositary receipts markets over multiple time periods. This assessment is based on the impact of ESG investment and fossil-fuel divestment practices on portfolio behavior in terms of risk, return, stability, diversification, and climate mitigation credentials of associated investment decisions. We highlight two applications of DivFolio. The first revolves around using sector scanning to divest from a specialized portfolio featuring constituents of the FTSE 100. The second, rooted in actuarial considerations, focuses on divestment strategies informed by environmental risk assessments for mixed pension portfolios in the US and UK.

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.

This paper introduces gemact, a Python package for actuarial modeling based on the collective risk model. The library supports applications to risk costing and risk transfer, loss aggregation, and loss reserving. We add new probability distributions to those available in scipy, including the (a, b, 0) and (a, b, 1) discrete distributions, copulas of the Archimedean family, the Gaussian, the Student t and the Fundamental copulas. We provide an implementation of the AEP algorithm for calculating the cumulative distribution function of the sum of dependent, nonnegative random variables, given their dependency structure specified with a copula. The theoretical framework is introduced at the beginning of each section to give the reader with a sufficient understanding of the underlying actuarial models.