In this bicentenary year of Gompertz (1825) [23] we advance a conjecture; that the split between Newtonian and Leibnizian forms of calculus occurring in the eighteenth century had a long-term influence on actuarial science in the nineteenth and twentieth centuries. When the growing life insurance industry in Britain needed mathematical expertise, a profession with Newtonian roots was created to supply it, while elsewhere in Europe it was found in universities with Leibnizian roots. We consider the consequences up to the present day, when analysis in the Leibnizian branch has led via Kolmogorov [49] to modern financial mathematics.
Survival models based on individual lifetimes are a cornerstone of modern medical statistics. The foundations of survival analysis were laid by actuaries, driven by practical as well as theoretical benefits. However, technological limitations led the actuarial profession to leave the development of survival modelling to statisticians many decades later. This paper seeks to remind actuaries of their early leading role in this field, and perhaps to rekindle interest in what their forebears started, especially since computing resource has long ceased to be a limiting factor. This paper may also be of interest to non-actuaries in understanding the unique characteristics of actuarial data sets and of actuarial modelling requirements.
We examine climate-related exposure within a large credit portfolio, addressing transition and physical risks. We design a modeling methodology that begins with the Shared Socioeconomic Pathways (SSP) scenarios and ends with describing the losses of a portfolio of obligors. The SSP scenarios impact the physical risk of each obligor via a DICE-inspired damage function and their transition risk through production, requiring optimal adjustment. To achieve optimal production, the obligor optimizes various energy sources to align its greenhouse gas (GHG) emission trajectories with SSP objectives, while accounting for uncertainties in consumption trajectories. Ultimately, we obtain a Gaussian factor model whose dimension is of the order of the number of obligors. Two efficient dimension reduction methods (Polynomial Chaos Expansion and Principal Component Analysis) provide a fast and accurate method for analyzing credit portfolio losses.
Urban flood insurance now relies heavily on catastrophe models that convert physical hazards into insured losses. For floods, this conversion is never direct: rainfall becomes runoff, runoff becomes local inundation, inundation becomes damage, and damage becomes claims. Each step depends on hydrological, hydraulic and contractual choices. Here we treat hydrological equifinality (the coexistence of several plausible models consistent with the same observations) as a source of actuarial model risk. The scope is deliberately urban: drainage networks, sewer–surface interactions, micro-topography, flood defences and concentrated insured values make model choices especially visible on the actuarial balance sheet. The relevant uncertainty is not only around a flood quantile, but around the model chain through which water becomes loss. Alternative credible representations of the same urban flood system can change premiums, underwriting classifications, reinsurance recoveries, capital requirements and the estimated value of prevention. Flood model outputs should therefore be accompanied by multi-model comparison, robust stress tests and transparent sensitivity analysis. Flood insurance should price floods, but it should also recognize the uncertainty of the models through which floods become insured losses.
Since the Lee and Carter [57] model, many researchers have focused on forecasting age-specific mortality rates. Recent research highlights the use of mortality improvements to increase forecasting accuracy. We propose a mortality improvement model inspired by interest rate modeling, where the changes in the mortality curve depend linearly on a reduced number of age-specific or “key” age(s). These variables are directly observable and capture the general mortality trend. Previous versions used one key age, whereas we introduce a second factor to improve the explanatory and forecasting power. Across populations, the first key age is consistently around 85, while the second is near age 30. We compare the accuracy of the models with seven benchmark mortality models in six populations, including the pandemic period. The results show that including a second factor enhances performance only when it is correlated with a sufficiently wide range of age-specific rates.
Reinsurance treaties are one of the main instruments used by insurance companies for reducing their risks and balancing their technical performance. The selection of Pareto-efficient reinsurance strategies under Solvency II involves a computationally intensive multi-objective optimization problem with mixed continuous and discrete variables. Traditional simulation-based methods, like grid-search or random search, often become intractable due to the high-dimensional search space and the cost of extensive simulations. While evolutionary algorithms are typically able to solve these problems, their computational cost often prohibits real-time decision-making and sensitivity analysis. In this paper, we propose a novel deep learning architecture that formulates the optimization task as an inverse design problem. Our framework couples a proxy network, acting as a differentiable surrogate for the insurer’s internal model, with a generator network that directly maps target profitability levels to optimal reinsurance structures. We employ Gumbel-Softmax relaxation to effectively optimize discrete treaty features, such as the reinsurance selection based on its credit quality step, within a fully differentiable pipeline. We introduce a deep active learning loop that iteratively refines the model’s accuracy in the Pareto-optimal region, minimizing the required simulation budget. We validate the framework through a numerical application on a multi-line non-life insurer calibrated to the Italian market, aiming to jointly maximize the Return on Equity and Solvency Ratio. Our results demonstrate that the proposed approach outperforms traditional simulation-based methods in terms of frontier determination. Moreover, it achieves statistical equivalence with the evolutionary benchmark, Non-Dominated Sorting Genetic Algorithm II (NSGA-II), while reducing the computational budget by approximately 75
In this paper, we conduct numerical experiments to evaluate decarbonization strategies for pension plans using a regime-switching multivariate normal return framework. Our approach distinguishes between climate dynamics and portfolio strategy. The climate dynamics are modeled as a regime-switching process, where the regimes represent exogenous climate risk mitigation strategies. Within each regime, we model sector-specific returns, which incorporate sector and regime dependent effects of transition and physical climate risks. These adjustments are educed from historical data and climate science literature. The portfolio decision concerns the pension plan’s decarbonization strategy. We simulate 30-year wealth paths under three decarbonization strategies. Our results show that the decision to decarbonize consistently outperforms inaction across the climate scenarios considered. These findings persist when investor behavioral biases, including recency bias and climate skepticism, are incorporated. However, significant tail risk persists across all strategies, reflecting the potential for systemic market failure under adverse climate outcomes. Overall, the analysis suggests that pension plan fiduciaries can pursue decarbonization not despite their financial obligations but in alignment with them, provided climate risks are properly integrated into long-term investment modeling.
Insurance products often cover substantial claims arising from diverse sources. To accurately model these losses, actuarial models must account for high-severity claims. An effective approach is to use a mixture model that fits a distribution to losses below a certain threshold, while modelling excess losses using extreme value theory. However, selecting an appropriate threshold remains a key challenge, as existing methods are sensitive to this choice and lack a universally accepted criterion. Bayesian Model Averaging (BMA) offers a promising solution by allowing the simultaneous consideration of multiple thresholds. In this paper, we demonstrate that an error integration BMA algorithm provides a flexible framework accounting for threshold uncertainty by combining models across multiple candidate values. This approach improves model accuracy by capturing the full loss distribution while mitigating sensitivity to any single threshold choice. When interpretability or tail-specific inference is needed, the method can also identify the most likely threshold supported by the data. We illustrate the usefulness of the proposed framework through simulation studies and an application to an automobile claims dataset from a Canadian insurer. We also examine a setting without predictive variables and compare our method to conventional threshold selection procedures based on goodness-of-fit tests applied to an actuarial dataset.
We investigate how rental insurance deductible choices and rent affordability jointly affect demand for rental insurance and the value of risk reduction for hurricane-related power outages and property damages using discrete choice experiments. We tested five rental insurance deductible menus with increasing (4–8) number of choice options and maximum deductible (2500–10,000). Our experiment was administered to a random sample of multi-family renters across 11 states and data analyzed with mixed logit models. Our analysis reveals high preferences for low (250 and500) deductible plans and a high likelihood of a zero willingness to pay (WTP) for standby power generators and windstorm-resistant buildings. However, propensity to choose lower deductibles decreases non-linearly with an increase in the coefficient of absolute risk aversion (CARA), and WTP for power outage or property damage risk reduction increases with CARA and decreases as the percentage of income spent on rent increases. Overall, renters with higher levels of deductibles are more likely to pay for hurricane risk reduction actions. We show that an optimal range and number of deductibles exist that maximizes full coverage and minimizes under-insurance, and that a mandatory rental insurance program without constraints on the deductible range could simultaneously increase insurance take-up and underinsurance, muting the desired effects of the program.
Biological age (BA) offers a promising approach for encapsulating complex health information into a single interpretable metric. This study evaluates BA methods as tools for prevention in insurance, focusing on their ability to predict mortality and disease incidence. Using National Health and Nutrition Examination Survey (NHANES) data, we compare five BA calculation methods—multiple linear regression (MLR), Klemera-Doubal Method (KDM), PhenoAge, calibrated PhenoAge, and Random Forest (RF). We include a practical application of estimating death counts from life tables. Our findings reveal that RF and calibrated PhenoAge consistently outperform other methods in mortality prediction and more accurately estimate observed death counts. While MLR and KDM lag in predictive performance, they demonstrate interpretability that may be valuable for some applications. PhenoAge showed the greatest flexibility and adaptability for prevention-focused applications, particularly for estimating death counts. However, a key challenge remains in calibrating BA methods to align with absolute mortality risks, as highlighted by their initial biases in estimating death counts. We argue that BA’s primary value lies in its dual role: a reliable risk estimator and an effective communication tool for promoting preventive health behaviors. By addressing calibration issues and tailoring BA methods to specific insurance contexts, this research underscores BA’s potential to improve prevention programs, aligning health incentives for both policyholders and insurers.
The drawdown of a stochastic process is the absolute distance to its historical peak. It is a widely used risk and performance measure in financial applications. For a diffusion process subject to a continuous-time control process, we consider a stochastic control problem targeting the simultaneous maximisation of growth of the running maximum and minimisation of the weighted occupation time in the area bounded away from it by at least d>0 . The model we consider corresponds to a diffusion risk model under proportional reinsurance. The optimal reinsurance strategies obtained lead to a stabilisation of this surplus process close to its own running maximum. In particular, they promote growth of the surplus while simultaneously avoiding large negative deviations from the current record high. By exploiting connections to Hamilton–Jacobi–Bellman-equations and reflected SDEs, we find explicit expressions for the value functions and strategies and show that the processes under the optimal feedback controls exist. We discuss examples and implications of our results in the context of the application.
Climate change brings serious challenges for various sectors of the economy, including the insurance industry. One important impact is its influence on mortality rates, which directly affect the financial reserves of life insurers. Accurately assessing this relationship is essential for ensuring the sustainability and solvency of life insurance portfolios. In this study, we present a framework for incorporating climate risk into the stress testing of life insurers’ reserves. Using temperature projections generated from a stochastic version of a widely used cost–benefit integrated assessment model, we simulate shocks to mortality rates driven by changes in global average temperatures, and evaluate their impact on pricing and reserving for life insurance and life annuity policies. Our numerical analysis demonstrates material risks of mispricing premiums when climate risk is ignored, and reveals substantial differences in reserve requirements, thereby highlighting the importance of integrating climate change considerations into actuarial modeling and financial planning for life insurers.
This paper introduces a unified framework of mixed Poisson spatio-temporal regression models for climate-related property insurance claims. Our approach integrates two model specifications which have been studied separately in the literature, a spatio-temporal (SP) model that explicitly accounts for spatial autocorrelation, and a temporal Besag model that leverages spatial random effects to smooth regional variations. For expository purposes, the spatio-temporal Negative Binomial (SP-NB) and temporal Besag Negative Binomial (Besag-NB) regression models are fitted to claim data related to flood and flood–windstorm events from a Greek property insurance company over the period 2012–2022. Parameter estimation is performed using an Expectation–Maximization algorithm for the SP-NB model and Integrated Nested Laplace Approximations for the Besag-NB model. Finally, the a posteriori (bonus–malus) premium rates derived from these models incorporate property-specific characteristics, geographical information, regional trends, individual experiences, and a flood vulnerability index that accurately reflects true exposure in flood-prone areas.
This paper introduces a model designed to support green transition and climate change adaptation, focusing on the role of insurance companies and public authorities. As climate change accelerates, climate risks are seen as uninsurable, unless alternative risk transfer methods are employed. One such method is the issuance of Catastrophe (CAT) bonds, which allow insurers to transfer risks to financial investors. However, these instruments alone do not guarantee a reduction in climate risk or foster a green transition. Our model assumes that firms (policyholders) are exposed to catastrophic risks, which can be mitigated by adopting green technologies (which we assume to include, by extension, any form of climate change adaptation). To encourage this transition, insurance companies, with support from a public authority, periodically issue resilience bonds, similar to CAT bonds. If a sufficient number of “virtuous" firms adopt green technologies, the risk - and thus interest rates - on these bonds decrease, allowing the bonds to finance the green transition, such as offering premium discounts to the adopting firms. This creates a dynamic interaction between bond rates and the proportion of firms using green technologies.The model outlines two scenarios: one where all trajectories converge to an optimal equilibrium (where all firms adopt green technologies and bond rates are minimal), and another where a sub-optimal equilibrium occurs with fewer firms adopting green technologies and higher bond rates. The paper’s main contribution is the development of a quantitative model for a green transition supported by financial instruments and public intervention, with a specific application to mitigating flood risk in Italy.
Consider a financial system comprising multiple individual companies. These companies jointly face a sequence of claim vectors arriving according to a Poisson process. Suppose that these companies make both risk-free and risky investments, with overall returns modeled by a geometric Brownian motion. In this dynamic, multidimensional setting, we introduce several systemic risk measures and conduct an asymptotic analysis of them. For claims that are heavy-tailed and either asymptotically dependent or asymptotically independent, we derive precise asymptotic formulas for the systemic risk measures. Numerical studies are carried out to evaluate the accuracy of these estimates, with particular emphasis on the roles of marginal tails and tail dependence.
We introduce , the Internal Risk Model of an artificial life insurer, designed to allow an easy benchmarking of nested simulation techniques for Solvency Capital Requirement ( SCR ) estimation under Solvency II and other actuarial methods. integrates an economic scenario generator and a cash flow projection model, enabling the computation of the available capital (basic own funds) through both the direct and indirect method. Leveraging a two-factor Gaussian model for stochastic short rates and a generalized Black-Scholes model for stock dynamics, the framework supports policyholder investments via guaranteed minimum-income benefit contracts. We extend the asset-liability management model by Diehl et al. (EAJ 13(1), 2022), and prove the theoretical convergence of the direct and indirect method under appropriate assumptions. Calibrated using interest rate caps from 2016 to the end of 2023, allows estimation of available capital distributions and SCR dynamics for each trading day in that range. The source code of written in MATLAB is publicly available on gitlab at https://gitlab.cc-asp.fraunhofer.de/itwm-fm-lv-public/openirm . We also provide standalone executables that, after installation, can be accessed via the command line interface or with the provided wrappers in R, Python and MATLAB.
Index insurance is often proposed to reduce protection gaps, especially for emerging risks. Unlike traditional insurance, it bases compensation on a measurable index, enabling faster payouts and lower claim management costs. This approach benefits both policyholders, through quick payments, and insurers, through reduced costs and better risk control due to reliable data and robust statistical estimates. An important difference with the concept of Cat Bonds is that the feasibility of such coverage relies on the possibility of mutualization. Mutualization, in turn, is achieved only if a sufficiently high number of policyholders agree to subscribe. The purpose of this paper is to introduce a model for the demand for index insurance and to provide conditions under which the solvency of the portfolio is achieved. From these conditions, we deduce a product that combines index and traditional indemnity insurance in order to benefit from the best of both approaches. We illustrate our results with a practical example involving the design of an index insurance product in the field of cyber insurance.
This article demonstrates the transformative impact of Generative AI (GenAI) on actuarial science, illustrated by four implemented case studies. It begins with a historical overview of AI, tracing its evolution from early neural networks to modern GenAI technologies. The first case study shows how Large Language Models (LLMs) improve claims cost prediction by deriving significant features from unstructured textual data, significantly reducing prediction errors in the underlying machine learning task. In the second case study, we explore the automation of market comparisons using the GenAI concept of Retrieval-Augmented Generation to identify and process relevant information from documents. A third case study highlights the capabilities of fine-tuned vision-enabled LLMs in classifying car damage types and extracting contextual information. The fourth case study presents a multi-agent system that autonomously analyzes data from a given dataset and generates a corresponding report detailing the key findings. In addition to these case studies, we outline further potential applications of GenAI in the insurance industry, such as the automation of claims processing and fraud detection, and the verification of document compliance with internal or external policies. Finally, we discuss challenges and considerations associated with the use of GenAI, covering regulatory issues, ethical concerns, and technical limitations, among others.
Payments in parametric insurance are linked to an index and thus decoupled from policyholders’ true losses. While this principle has appealing operational benefits compared to indemnity coverage, i.e. being efficient and cost effective, a downside is the discrepancy between payouts and actual damage, called basis risk. We show that in an asymmetrically weighted mean square error framework, the basis risk-minimizing payment schemes for pure parametric and parametric index insurance contracts can be expressed as conditional expectiles of policyholders’ true loss given a compensation-triggering incident. We provide connections to stochastic orderings and demonstrate that regression approaches allow easy implementation in practice. The results are visualized in parametric coverage for cyber risks and agricultural insurance.
With the rapid development of machine learning and deep learning techniques, actuaries and the broader insurance industry face a persistent trade-off between predictive accuracy and interpretability. This paper provides a comprehensive applied assessment of Explainable Boosting Machines (EBM) in a car insurance framework, focusing on claim frequency and severity modeling. EBM combines the additive structure of generalized additive models (GAM) with a cyclic gradient boosting algorithm, resulting in a glass-box model whose predictions are interpretable by design. Using real-world data, we empirically illustrate its practical relevance and compare EBM with modern benchmark models used in non-life insurance pricing. The evaluation considers (i) out-of-sample predictive accuracy, including Murphy diagrams and Bregman dominance tests, and (ii) calibration assessment using T-reliability diagrams and Murphy's score decomposition. Finally, we highlight the link between EBM predictions and Shapley values, showing how predictions can be transparently decomposed into exact main and pairwise interaction effects, providing actionable insights beyond predictive performance.
