Climate change is likely to increase the frequency, severity, and duration of heat waves in many countries. To plan mitigation, adaptation, and resilience strategies, it is necessary to quantify heat wave risk at both the local level and the country level. A new, more granular methodology is proposed in order to integrate the impact of heat waves in hexagonal France on mortality with a short-term stress scenario. Based on open data and reproducible methodology, the approach can be used as a starting point to investigate other effects, such as urban heat islands. The present application is based on in situ observational weather data and environmental vulnerability data to construct adapted geographical clusters without relying on the administrative division of the territory. Excess mortality is modeled as a function of weather using machine learning. Using recent knowledge of climatology, we construct extreme weather scenarios to calculate a shock to mortality. Short-term shocks are compared, and their respective merits are discussed. The methodology has been shown to generate mortality shocks up to five times greater than those estimated by the French regulatory authority.
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.
Accurate and internally coherent crop-yield forecasts are important for agricultural risk management, crop-insurance ratemaking, and regional risk assessment under climate variability. However, crop yields are influenced by high-dimensional and strongly correlated weather conditions, while forecasts produced at different spatial levels often violate aggregation constraints. Existing studies focus on yield prediction within individual regions and pay limited attention to weather-informed forecasting, hierarchical coherence, and insurance-oriented risk measurement. This paper develops an integrated framework for hierarchical crop-yield forecasting and risk assessment by combining dimensionality reduction for high-dimensional weather variables, probabilistic forecasting, and forecast reconciliation. Using county- and state-level spring and winter wheat yields in Montana from 1982 to 2022, we compare alternative base forecasting models and reconciliation methods under scenarios with and without weather information. Forecast performance is evaluated using point and probabilistic scoring rules, and the reconciled predictive distributions are used to construct scenario-based measures of downside yield risk. The results show that incorporating weather information and hierarchical reconciliation improves the quality and coherence of hierarchical yield forecasts. The resulting probabilistic forecasts provide a basis for loss-rate estimation, cross-county risk comparison, and spatial risk mapping and also support crop-insurance ratemaking under a retain-cede game between private insurers and the government.
A crucial issue in a dynamic framework is how risk valuations at different times are interrelated. In this regard, the notion of time consistency was widely introduced and discussed in the literature. A time-consistent dynamic valuation states that a future payoff preferred to another payoff at some future time point should already be preferred to this payoff today. This paper aims to construct a time-consistent, dynamic version of the Three-step method introduced in Deelstra et al. ((2020). ASTIN Bulletin: The Journal of the IAA, 50(3), 709-742.) for hybrid life Pure Endowment products, employing a backward iteration scheme. The backward scheme is illustrated in a dual-iteration approach using a Pure Endowment product without profit sharing. Furthermore, we explore the continuous-time limit of the backward scheme, incorporating profit-sharing into the Pure Endowment to investigate a hybrid life payoff. Our analysis demonstrates that the presence of the diversifiable component undermines the time-consistency of the dynamic three-step method. Consequently, the time-consistent price of the actuarial part shows a notable increase. To address this, and in accordance with Devolder and Leb & egrave;gue ((2016). Risks, 4(4), 49.), we present a reduced time-consistent variant by decreasing the safety loads in each iterative step of the backward scheme.
We develop a decision-support framework for cyber risk mitigation policies from the perspective of an organization with limited resources for security controls, upgrades, and cyber insurance. To balance the conflicting optimization objectives of the organization and the insurer, we propose a bi-level model that endogenously derives optimal strategies for both parties, accounting for key uncertainties underlying a cyber attack. We find that cyber insurance coverage increases with premium size, though this depends on the effectiveness of system upgrades. Notably, the latter has an ambiguous impact on the equilibrium budget allocation strategy and insurance contract design, such that a more effective upgrade need not attract a commensurately larger budget allocation. We further show that information asymmetry regarding the insurer's risk aversion can lead the defender to a suboptimal budget allocation, resulting in higher realized losses relative to the symmetric-information benchmark.
Insurance risk arising from natural catastrophes such as earthquakes is a key component of the minimum capital test for federally regulated property and casualty insurance companies. This paper proposes an integrated, open-source, simulation-based actuarial framework for the assessment of earthquake insurance risk and solvency capital requirements. The framework combines spatio-temporal earthquake occurrence modeling, physics-informed ground-shaking estimation based on Canadian seismic hazard maps, building exposure and vulnerability modeling, and detailed insurance loss and claim calculations within a unified pipeline. Spatial heterogeneity in seismic risk is captured through kernel-based spatio-temporal point process modeling, while Voronoi-based deviance residuals are employed as localized diagnostic tools to validate model adequacy. Simulated insured losses are used to estimate regional and country-wide probable maximum losses (PMLs), and a new capital aggregation formula is proposed that explicitly incorporates cross-provincial dependence in earthquake losses, in contrast to the current region-based regulatory aggregation. The proposed framework enables spatially resolved loss and capital assessment at a fine geographic scale and is implemented in a fully reproducible open-source environment. An interactive web application is also provided to allow users to simulate earthquake damage and the resulting financial losses and insurance claims at user-specified epicenter locations.
This paper assesses the impact of demographic risk on a portfolio of equity-linked insurance contracts featuring a Cliquet-style guarantee, in which the policyholder accrues, on an annual basis, interest equal to the maximum between the return on a risky portfolio and a guaranteed minimum rate. We provide closed-form expressions for inflows, outflows, and reserves for such a portfolio through a cohort-based approach. In accordance with market-consistent actuarial principles, we determine both the no-arbitrage value of the liabilities and the structure of the hedging portfolio that replicates the guaranteed benefits. We quantify demographic risk by separately assessing the capital requirements for both idiosyncratic and trend risks. The capital requirement is computed over a one-year horizon using a 99.5% Value-at-Risk measure, consistent with the Solvency II regulatory framework. The model accommodates different regulatory contexts, allowing for jurisdiction-specific rules and accounting standards. Numerical simulations highlight how the portfolio's risk profile is affected by demographic volatility, which is influenced by policyholder age, policy duration, and dispersion of the sums insured. Additionally, trend risk depends on both mortality volatility and the specification of the longevity model. This framework supports insurers in evaluating, hedging, and managing demographic risk in market-linked life insurance products.
This paper addresses the gap between theoretical modeling of cyber risk propagation and empirical analysis of loss characteristics by introducing a novel approach that integrates both approaches. We model the development of cyber loss counts over time using a discrete-time susceptible-infected-recovered process, linking these counts to covariates, and modeling loss severity with regression models. By incorporating temporal and covariate-dependent transition rates, we eliminate the scaling effect of population size on infection counts, revealing the true underlying dynamics. Simulations show that this susceptible-infected-recovered framework significantly improves aggregate loss prediction accuracy, providing a more effective and practical tool for actuarial assessments and risk management in the cyber risk context.
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.
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.
Pension fund populations often have mortality experiences that are substantially different from the national benchmark. In a motivating case study of Brazilian corporate pension funds, pensioners are observed to have mortality that is 40-55% below the national average, due to the underlying socioeconomic disparities. Direct analysis of a pension fund population is challenging due to very sparse data, with age-specific annual death counts often in low single digits. We design and study a collection of stochastic subpopulation frameworks that coherently capture and project pensioner mortality rates via deflator factors relative to a reference population. Superseding parametric approaches, we propose Gaussian process (GP)-based models that flexibly estimate age- and/or year-specific deflators. We demonstrate that the GP models achieve better goodness of fit and uncertainty quantification. Our models are illustrated on two Brazilian pension funds in the context of exogenous national mortality tables. The GP models are implemented in R Stan using a fully Bayesian approach and take into account over-dispersion relative to the Poisson likelihood.
We develop a continuous-time stochastic model for optimal cybersecurity investment under the threat of cyberattacks. The arrival of attacks is modeled using a Hawkes process, capturing the empirically relevant feature of clustering in cyberattacks. Extending the Gordon-Loeb model, each attack may result in a breach, with breach probability depending on the system's vulnerability. We aim at determining the optimal cybersecurity investment to reduce vulnerability. The problem is cast as a two-dimensional Markovian stochastic optimal control problem and solved using dynamic programming methods. Numerical results illustrate how accounting for attack clustering leads to more responsive and effective investment policies, offering significant improvements over static and Poisson-based benchmark strategies. Our findings underscore the value of incorporating realistic threat dynamics into cybersecurity risk management.
In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsurance company with dynamic contagion claims. To deal with conventional and emerging catastrophic events, we propose the use of a compound dynamic contagion process for the catastrophic component of the liability. Under the premise that there is an absence of arbitrage opportunity in the market, we obtain arbitrage-free premiums for these contracts. To this end, the Esscher transform is adopted to specify an equivalent martingale probability measure. We show that reinsurers have various ways of levying the security loading on the net premiums to quantify the catastrophic liability in light of the growing challenges posed by emerging risks arising from climate change, cyberattacks, and pandemics. We numerically compare arbitrage-free catastrophe stop-loss reinsurance premiums via the Monte Carlo simulation method. We also compare them with those from generalized compound Hawkes/compound Cox cases. Sensitivity analyses are performed by changing the retention level, the Esscher parameters, and the intensity parameters.
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 model and functional regression formulations applied to 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 dynamic Nelson-Siegel model. We conducted the stress testing analysis of yield curves term-structure within a dual economy framework. The bond ladder portfolio was examined through a case study focused on spread modelling using historical data for US Treasury and UK bonds.
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.
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.
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.
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.
