Abstract Calendar year effects introduce dependence in loss development data and challenge traditional reserving assumptions. This paper proposes an extension of the log-normal Mack chain ladder model, which closely aligns with Mack’s assumptions, by incorporating external information on the relative impact of calendar year effects. Parameter estimation is performed via an EM algorithm, and standard errors are derived. The approach enables a principled integration of market information and is illustrated using industry data exhibiting pronounced calendar year effects.
High-frequency mortality data have attracted growing attention, but their use has largely been confined to specific applications rather than general modeling and forecasting. Such data pose new challenges to traditional mortality models due to pronounced seasonal patterns and short-term fluctuations. To address these challenges and produce more accurate forecasts with the high-frequency mortality data, this paper introduces a novel integration of gradient boosting techniques into traditional stochastic mortality models under a multi-population setting. Our key innovation lies in using the Li and Lee model as the weak learner within the gradient boosting framework, replacing conventional decision trees. Empirical studies are conducted using weekly mortality data from 30 countries (Human Mortality Database, 2015-2019). Empirical evidence highlights that the proposed methodology not only enhances model fit by accurately capturing underlying mortality trends and seasonal patterns but also achieves superior forecast accuracy, compared to the benchmark models. We also investigate a key challenge in multi-population mortality modeling: how to select appropriate subpopulations with sufficiently similar mortality experiences. A comprehensive clustering exercise is conducted based on mortality improvement rates and seasonal strength. The empirical results demonstrate that our proposed model maintains strong forecast accuracy across different clustering configurations, thereby reducing the need for extensive data preprocessing.
We propose a matrix-based factor analysis model for predicting the probability of insurance claims. The model employs projected principal component analysis (PPCA), which enhances the estimation of unobserved latent factors by projecting a data matrix onto a linear space spanned by insured-specific features. This approach addresses the overparameterization problem when the number of insured-specific features and insurance coverages is large, enabling more accurate estimation of claim probability than conventional methods. Using a large-scale health insurance dataset from a leading life insurer in South Korea, we demonstrate that the proposed model outperforms conventional and machine-learning benchmarks, such as logistic regression and XGBoost, in predicting claim probabilities. We further determine that our model can reduce computational time by approximately 86% and 98% compared to logistic regression and XGBoost, respectively. The proposed model provides a unified and scalable framework for modeling high-dimensional claim probabilities, offering practical value for underwriting, risk management, and personalized insurance product design.
Recent advances in healthcare and rising life expectancy intensify longevity risk, motivating a deeper understanding of how cause-of-death (COD) rates interact. Using male COD data from 1978 to 2018 in the United States, we develop a copula-based hierarchical framework for seven major causes: cancer, diabetes, external causes, influenza, mental disorders, nephritis, and vascular disease. The framework integrates reconciliation, hierarchical dependence, and long-run equilibrium using a Lee-Carter (LC) setting. More specifically, the LC period indices are estimated under reconciliation penalties and are modeled through a sparse vector error correction model, with dependence captured by a hierarchical Archimedean copula. Two applications illustrate the value of our approach. In out-of-sample forecasting, the framework outperforms the standard LC model by improving the accuracy of aggregate mortality rates. In structural analysis, fitted connectedness reveals that diabetes and vascular disease act as net transmitters of mortality shocks, while cancer and external causes are net receivers. These insights help actuaries, demographers, clinicians, and policymakers enhance mortality forecasting to assess whether prioritizing government interventions for high-transmission causes could potentially maximize overall mortality improvements for society.
In classical credibility theory, estimation is typically limited to the hypothetical mean, restricting its use for premium principles that depend on higher-order moments. To address this, we develop a credibility-based framework for estimating the process variance under both known and unknown hypothetical means and apply these estimators to a broad class of variance-related premium principles, including the expected value, variance, standard deviation, and modified-variance principles. The estimators are derived via constrained linear projection techniques, minimizing the mean squared error between the estimator and the true process variance. Explicit formulas are obtained that are optimal among affine transformations of the data. The proposed estimators exhibit desirable statistical properties, including conditional unbiasedness, consistency, mean squared error convergence, and asymptotic normality. Numerical studies demonstrate their favorable convergence behavior, and an empirical analysis with real insurance data highlights their practical relevance. This framework extends B & uuml;hlmann's classical credibility theory to second-moment estimation while remaining computationally tractable and requiring only mild moment conditions, without specifying the population or prior distributions.
This paper extends the traditional group self-annuitisation framework by explicitly incorporating mortality heterogeneity among participants. Heterogeneity stems from multiple factors that lead individuals to age at different paces, despite being born in the same year. Ageing is modelled as a finite-state continuous-time Markov process where each state represents a distinct phase of physiological deterioration, and transitions capture the stochastic progression towards death. Benefits are differentiated by ageing state and, after issue, they are dynamically adjusted in response to the realised evolution of both ageing and mortality. Our design is novel in its use of the Markov ageing framework within a risk-sharing scheme and in how benefits are updated. Indeed, both benefits and their respective adjustment coefficients are state-specific. Through the explicit modelling of cross-subsidies across states, the design ensures that actuarial equivalence between benefits and available resources is preserved both at the pool level and within each ageing state. However, we find that benefit adjustments based on actuarial equivalence may display undesirable patterns in some ageing classes, when their size shrinks substantially; this happens, in particular, in the younger ageing states, which are likely to empty out. To contrast such effects, we introduce a design preserving a target level of differentiation across states that mitigates the unfavourable impact of a declining size for younger ages. In our analysis, we point out that such a design (which is desirable in many respects) implies solidarity effects across states. Such effects can be identified by comparing benefit amounts under the two assumptions (i.e., benefits adjusted according to actuarial equivalence or so to preserve a predefined level of differentiation). The proposed framework is tested using Australian mortality data.
This study investigates a hybrid variable annuity (VA) contract that combines guaranteed minimum accumulation benefit (GMAB) and guaranteed minimum death benefit (GMDB) riders, with the added flexibility for policyholders to surrender prior to maturity. The contract guarantees the return of premiums or a greater rolled-up value at either maturity or death. We propose a novel two-account structure: an investment account tied to the underlying fund, from which management fees are deducted, and a separate cash account for the deduction of insurance fees for funding the GMAB and GMDB riders. This design generalizes the conventional single-account model by decoupling fee sources. From the policyholder's perspective, we derive actuarially fair insurance charges and show that the two-account framework delivers substantially lower guarantee fees compared to the classical design, improves contract characteristics by reducing the effective moneyness of embedded guarantees, thereby discouraging early surrenders and mitigating mortality risk mispricing. Furthermore, we show that bundling survival and death benefits results in higher fair fees than the sum of standalone riders, thereby enhancing the product's appeal to insurers. The analysis also incorporates taxation considerations up to a predefined preservation age, reflecting regulatory and practical product design constraints.
In order to mitigate a life annuity provider's (insurer's) longevity risk exposure, we propose a general longevity risk transfer policy between the insurer and a reinsurer. The reinsurance premium is calculated according to the expected premium principle. Under an expected utility maximization framework, we apply the variational method to derive the necessary and sufficient condition for the optimal longevity risk transfer policy. We find that the optimal strategy takes the form of excess of age policy, which means that the insurer is only liable for the benefit payment up to the optimal deductible age, and the remaining benefit payment is covered by the reinsurer. Furthermore, we assess the viability of the reinsurer underwriting the optimal longevity risk transfer policy. Numerical examples show that the optimal longevity risk transfer policy can effectively improve the insurer's relative gains and reduce the insurer's longevity risk exposure.
In this paper, we investigate how policyholder information and broader economic conditions jointly influence the duration of credit life insurance contracts in the French market. Employing a proportional-intensities regression framework built on inhomogeneous phase-type distributions, we capture the way covariates shape the distribution of policy lifetime until a lapse occurs. The model is estimated via a specialized expectation-maximization algorithm, adapted to handle censored data, covariates, and feature selection through shrinkage. Our analysis of real-world data shows that different policyholder attributes and economic factors can significantly alter lapse behavior, with effects varying across insurance products, individuals, and economic cycles. These findings highlight the importance of integrating both individual-level and macroeconomic indicators in lapse risk assessment, ultimately informing more accurate pricing and allowing for improved risk management strategies.
Lifetime pension pools-also known as group self-annuitization plans, pooled annuity funds, and variable payment life annuities in the literature-offer retirees lifelong income by collectively managing mortality risk and adjusting benefits based on the investment performance and the mortality experience within the pool. The benefit structure hinges on two key design parameters: the investment policy and the hurdle rate. However, past research offers limited guidance on optimal asset allocation in such settings, often relying on overly simplistic strategies. Furthermore, the choice of hurdle rate has received virtually no attention in the literature. This study addresses this gap by jointly analyzing optimal hurdle rates and investment strategies using a dynamic programming approach that allows for varying degrees of risk aversion via a hyperbolic absolute risk aversion utility function. Our findings reveal that, as risk aversion increases, the model favours more conservative portfolios and lower hurdle rates; conversely, lower risk aversion supports riskier allocations and higher hurdle rates. The threshold parameter-which reflects the minimum acceptable level of consumption-plays a critical role in shaping the hurdle rate behaviour.
Global mortality rates continue to decline, and life expectancy continues its upward trend. Besides mortality levels, policymakers and providers of financial and health services would also be interested in disability prevalence and its potential future trajectories. The length of time in good health versus the duration with major disabilities or long-term illnesses has significant financial implications for both individuals and society. In this paper, we develop Bayesian common factor models to analyse Australian age- and sex-specific disability prevalence rates. In particular, there are one or more common factors shared by both sexes, as well as specific factors for each sex. Retirement villages are purpose-built residential complexes designed for relatively healthy retirees to live as neighbours and share a communal lifestyle. We apply the model forecasts and simulations to valuate a typical retirement village contract. The cost of this accommodation service is determined by the resident's total length of stay, which can be estimated using forecasted and simulated disability prevalence rates and mortality rates from our proposed models.
Consider a general mortality-linked security (MLS) with a bounded payoff contingent on the evolution of the underlying mortality rate and the performance of associated risky assets. The mortality rate and asset prices are assumed to jointly follow a multivariate It & ocirc; process, driven by both a multivariate Brownian motion and a Poisson point process. We follow the utility indifference approach to pricing this MLS under the physical measure. To this end, we employ backward stochastic differential equations (BSDEs) to characterize the optimal investment strategy and the value function for the involved optimization problems. We then solve the resulting nonlinear BSDEs with a non-Lipschitz generator. This methodology, which combines the utility indifference approach with BSDE techniques, provides numerical tractability through Monte Carlo simulations. Finally, we conduct comprehensive numerical studies on the valuation of several concrete MLSs, with a focus on the sensitivity analysis of the indifference prices against various key model parameters, including, in particular, the correlation between the underlying mortality rate and asset price.
Owing to their innovative guarantee features, the popularity of variable annuities has gained significant traction as suitable retirement products in recent years. Amongst these guarantees, the guaranteed minimum income benefit (GMIB) stands out as an appealing rider that can be integrated into variable annuity contracts. In this research, we construct a comprehensive modelling framework that encompasses three sources of uncertainty, namely interest risk, mortality risk and investment risk, with the aim of valuing the GMIB. These risk factors are modelled stochastically whilst accounting for the interdependence between interest and mortality risks. The num & eacute;raire transformation technique is utilised in our approach, capitalising on the concepts of the forward and endowment-risk-adjusted measures. By considering two distinct settings of the Benefit Base functions, we derive an analytic solution for the GMIB. Our numerical findings demonstrate the superiority of our proposed methodology vis-& aacute;-vis the standard Monte Carlo simulation as a benchmark in terms of computational accuracy and efficiency, achieving a remarkable average improvement of 99% computing time reduction compared to the benchmark. Furthermore, we conduct an extensive sensitivity analysis to explore the levels of impact of various model parameters on the value of the GMIB.
We propose a deep reinforcement learning (RL) framework designed to optimize the hedging of specific, user-defined risk factors-referred to as targeted risks-in financial instruments affected by multiple sources of uncertainty. Our methodology uses Shapley value decompositions to establish source of risk grouping's contribution to the projected contract cash flows, providing a clear attribution of the profit and loss to distinct risk categories. Leveraging this decomposition, we apply deep RL to hedge only the targeted risks, while leaving non-targeted risks mostly unaffected. In addition, we introduce a joint neural network architecture in which the agent network utilizes risk estimates from a risk measurement neural network to stabilize the hedging strategy, taking into account local risk dynamics. Numerical experiments show that our approach outperforms traditional methods, such as delta hedging and traditional deep hedging, significantly reducing targeted risks in variable annuities while maintaining flexibility for broader applications.
This paper develops a unified framework for catastrophe (CAT) bond pricing that integrates distortion operator theory with recurrent neural network (RNN) estimation. A novel peer-adjusted distortion factor is introduced, constructed from both the Wang transform and the jump-diffusion (JD) distortion operator, and calibrated using the market-weighted spread of comparable CAT bonds together with the target bond's expected loss. This factor embeds prevailing investor sentiment, reinsurance capacity, and market liquidity into the distortion measure, enabling consistent pricing inference even when the bond's own spread is unobserved. Empirically, the JD distortion model systematically outperforms both the canonical Wang transform and the raw expected loss in in-sample and out-of-sample tests, capturing discontinuous repricing and tail-risk compensation with greater precision. Extending the framework to a multifactor specification that combines actuarial fundamentals with financial-market covariates further enhances explanatory and predictive performance. From a methodological perspective, the RNN serves as a structural estimator for the parameters of the distortion operators, achieving higher accuracy, stability, and computational efficiency than conventional approaches such as MLE, GMM, or ensemble regressors. By unifying distortion operators with neural estimation, this study advances both the methodological and empirical foundations of CAT bond pricing within actuarial science.
Individual loss reserving methods have undergone substantial development in the past decade, driven by increased accessibility to granular-level insurance claims data. This paper presents a micro loss reserving model tailored for multi-coverage insurance policies, where a single insurance claim might trigger payments from multiple coverage types. We employ a copula-based multivariate regression approach to jointly model the settlement time and loss amount, effectively capturing the dependence among various types of loss amounts and their correlation with the settlement time. We stress the importance of considering both types of dependence for accurate reserving prediction and uncertainty quantification. Furthermore, we propose computationally efficient algorithms for parameter estimation and dynamic prediction. Through numerical experiments and real data analysis, we demonstrate the effectiveness of our proposed multivariate predictive model in loss reserving applications.
In this paper, we design a novel axiomatic approach to evaluating the joint risk of multiple insurance risks under dependence uncertainty. To be precise, we first establish a joint risk measure for non-negative multivariate risks, which we refer to as a (scalar) distortion joint risk measure. Then, we characterize it via a new set of axioms. Moreover, we introduce a new class of vector-valued distortion joint risk measures for non-negative multivariate risks and discuss their basic properties. Finally, comparisons with some existing vector-valued multivariate risk measures are made. It turns out that those vector-valued multivariate risk measures have forms of vector-valued distortion joint risk measures, respectively. This paper provides some relevant theoretical results about the evaluation of joint risk under dependence uncertainty.
We study Stackelberg Equilibria (Bowley optima) in a monopolistic centralized sequential-move insurance market, with a profit-maximizing insurer who sets premia using a distortion premium principle, and a single policyholder who seeks to minimize a distortion risk measure. We show that equilibria are characterized as follows: In equilibrium, the optimal indemnity function exhibits a layer-type structure, providing full insurance over any loss layer on which the policyholder is more pessimistic than the insurer's pricing functional about tail losses; and no insurance coverage over loss layers on which the policyholder is less pessimistic than the insurer's pricing functional about tail losses. In equilibrium, the optimal pricing distortion function is determined by the policyholder's degree of risk aversion, whereby prices never exceed the policyholder's marginal willingness to insure tail losses. Moreover, we show that both the insurance coverage and the insurer's expected profit increase with the policyholder's degree of risk aversion. Additionally, and echoing recent work in the literature, we show that equilibrium contracts are Pareto efficient, but they do not induce a welfare gain to the policyholder. Conversely, any Pareto-optimal contract that leaves no welfare gain to the policyholder can be obtained as an equilibrium contract. Finally, we consider a few examples of interest that recover some existing results in the literature as special cases of our analysis.
In machine learning-based mortality models, interpretation methods are well established, and they can reveal structures resembling the age or time effects in traditional mortality models. However, in the reverse direction, using such traditional components to guide the initialization of a neural network remains highly challenging due to information loss during model interpretation. This study addresses this gap by exploring how components from pre-fitted traditional mortality models can be used to initialize neural networks, enabling structural information to be incorporated into a deep learning framework. We introduce Kolmogorov–Arnold Networks (KAN) and first construct two shallow models, KAN[2,1] and ARIMAKAN, to examine their applicability to mortality modeling. We then extend the Combined Actuarial Neural Network (CANN) into a KAN-based Actuarial Neural Network (KANN), in which classical model components calibrated via generalized nonlinear models or generalized additive models are naturally used for initialization. Three KANN variants, namely KANN[2,1], KANNLC, and KANNAPC, are proposed. In these models, neural networks assist in improving the accuracy of traditional models and help refine the original parameter estimates. All KANN-based models can also produce smooth mortality curves as well as smooth age, period, and cohort effects through simple regularization. Experiments on 34 populations demonstrate that KAN-based approaches achieve stable performance while balancing interpretability, smoothness, and predictive accuracy.
Assessing systemic risk presents a significant challenge in finance and insurance, where conditional risk measures are essential for capturing contagion effects. This paper introduces two novel systemic risk measures – conditional interval value-at-risk (CoIVaR) and conditional interval expected shortfall (CoIES) – which extend traditional metrics by incorporating interval-based uncertainty. A formal theoretical framework is developed for both measures, offering a detailed characterization of their key properties and risk contributions. We then propose a comprehensive comparison methodology for systemic risk assessment, leveraging stochastic orders, dependence structures, and marginal distributions to establish conditions for ranking risk vectors. Finally, through numerical experiments and real-world stock market applications, we demonstrate the practical utility of CoIVaR and CoIES in quantifying systemic risk under uncertainty. The findings provide valuable insights into systemic risk propagation and establish a robust foundation for risk management in interconnected financial systems.
