Fairness adjustments in insurance pricing are defined relative to a reference population, i.e., to a joint distribution of (X, D, Y) where X are rating factors, Dare protected attributes, and Y is the claim amount. Because an insurer's portfolio is generally a selected subpopulation, portfolio and population reference distributions typically differ, so portfolio-calibrated and population-calibrated fairness adjustments need not coincide. In what follows, we use selection bias as an umbrella term for any discrepancy between an observed sample and its target population. We call portfolio composition bias the insurance-specific form of selection bias induced by the portfolio inclusion mechanism (underwriting/marketing), which makes each insurer's portfolio a selected subpopulation. Relying on causal inference and a portfolio composition indicator, we characterize how portfolio composition bias affects common premium adjustments (unawareness, discrimination-free pricing, and transport-based corrective pricing), and we provide restrictive conditions under which portfolio and population adjustments coincide. We propose estimators to recover the fairness-adjusted premiums on the regulator-intended target population from selection-biased data, by using externally available information on the population marginal distribution of the prohibited attribute D. We study this scope mismatch from the policyholder's perspective: we model the market premium faced by a newly entering policyholder (not yet assigned to any portfolio) as a mixture of insurer-specific premiums, weighted by the probability of being assigned to each insurer. Under this view, a pricing rule can satisfy a fairness criterion within each insurer's portfolio yet produce direct or proxy discrimination in the market when portfolio inclusion depends on X and/or D. Finally, we show that enforcing portfolio-level balance on population-intended fair premiums can reintroduce portfolio composition bias, highlighting a regulatory trade-off between portfolio balancing and market-wide fairness. We focus on recoverability: which population-level fairness targets are identifiable from portfolio data, and what minimal external information is required to recover them.
This paper examines the implications of incorporating education as a risk factor into life insurance, focusing on its effects on insurers' income statements. While the relationships between education and mortality have been widely studied in demography, epidemiology, and public health, their relevance for the insurance sector remains largely unexplored. To address this gap, we introduce the Education-based Risk Indices (ERI), which quantify differential mortality risk by educational level across ages and sexes relative to the population average, and analyse how the ERI can inform premium setting and shape competition in a market of rational insureds. The complete approach is illustrated with data from Spain. Individual-level population and mortality microdata from 2021 to 2022 are employed to construct life tables by education and derive the ERI by modelling the relationship between stratified and overall death rates. An application is presented through a simulation of a competitive insurance market in which one company adopts ERI-based pricing while another relies solely on age and sex. The findings indicate that integrating education into pricing strategies yields a measurable competitive advantage, enhancing insurers' financial performance relative to traditional approaches. Robustness analyses and a discussion about issues to be considered in real applications are also included.
This study applies the ruin theory to the government’s fiscal problem of infrastructure protection against a risk of destruction associated with reconstruction expenditure. In preparation for the possibility of large-scale expenditure, the government will take out an excess-of-loss (XL) insurance policy as an insured party and, at the same time, invest in physical disaster damage reduction facilities (i.e., “risk reduction’’) in advance. The government decides on the layers of risk to be covered by risk reduction, risk retention, and insurance, so as to maximize the expected value of fiscal resources, subject to the constraint that the probability of financial collapse (i.e., “ruin’’) is kept below a certain level. Furthermore, the study clarifies the composition of the impacts of risk reduction and insurance on ruin probability and numerically analyzes the optimization problem based on a hypothetical setting constructed using data from Vanuatu. The numerical example demonstrates a case in which the ruin probability rises sharply within specific short intervals of risk reduction and insurance attachment points, surpassing the constraint threshold, suggesting the need for careful selection of countermeasure combinations. It further clarifies that the lower the government’s initial financial reserves, the more extensive the insurance coverage should be.
Mortality data exhibit a hierarchical structure, where total death counts equal the sum of sex-specific death counts. While hierarchical forecasting reconciliation methods have improved mortality forecasts at aggregate levels, they typically address only the out-of-sample stage and overlook reconciliation during in-sample modelling. To bridge this gap, we propose three copula-based approaches within the standard Lee-Carter (LC) framework to reconcile total mortality rates using sex-specific rates. By incorporating reconciliation at the in-sample stage, these methods aim to improve parameter estimation and thereby enhance out-of-sample forecast accuracy. Using data from Australia, the United Kingdom, the United States, France, and Japan for ages 65-100 over the period 1950-2020, we demonstrate that our approaches outperform both the traditional LC model and LC-based hierarchical reconciliation. The proposed perfect reconciliation method, which achieves finite-sample reconciliation, consistently delivers the best performance across a range of sensitivity analyses. We further illustrate the practical utility of this integrated approach in forecasting life expectancy and pricing fixed-term annuities, highlighting its broader applicability in actuarial practice.
The growing elderly population has intensified interest in financial products such as reverse mortgages (RM) with long-term care (LTC) cost coverage, which provide funding solutions for senior homeowners facing income shortfalls and rising care costs. This study advances existing research on reverse mortgage contracts with long-term care cost coverage (RM_LTC) by systematically examining how the granularity of health state information influences contract valuation, particularly in the presence of prepayment options and market risks. We develop a numerical framework to determine the fair annuity rate of RM_LTC contracts. The framework jointly captures stochastic house prices, interest rates, and multi-state health transition matrices calibrated with real-world disability data. Our results show that finer health state granularity improves the precision of prepayment decisions, increasing the value of embedded options while lowering fair annuities. Conversely, relying on less detailed health information leads to an underestimation of option value and results in higher annuities offered to homeowners. By clarifying these mechanisms, the study underscores the pivotal role of health state granularity in valuing RM_LTC contracts and provides insights into the design of home-equity-based financing for long-term care.
This paper proposes a new metamodeling framework for valuing variable annuity (VA) portfolios that utilizes SHapley Additive exPlanations (SHAP) to select a representative sample in an explainable manner. When applying metamodeling techniques to principle-based VA portfolio valuation, selecting a representative sample with clear and explainable criteria is necessary. The proposed approach involves (i) training a surrogate neural network with existing valuations for the same portfolio across multiple market scenarios and (ii) decomposing the overall risk of a VA contract into clearly separated contributions from each risk driver using SHapley Additive exPlanations. This decomposition results in an informative and explainable representation of the VA contract data, which can later be used to select the representative sample under a new market scenario. Furthermore, by fine-tuning the surrogate neural network with the selected representative sample, the proposed method offers a systematic way to improve the neural network’s performance in the new market scenario. Our numerical analyses show that the proposed method outperforms conventional methods in the existing literature in prediction accuracy and goodness of fit.
This paper develops an economic framework for optimal longevity risk transfer between a buyer and a seller with different risk aversions. We compare static (long-dated, pre-committed) and dynamic (short-dated, rolled) longevity swaps in a Stackelberg game. We find that static contracts are preferred when the buyer is more risk averse, while dynamic contracts are preferred when the seller is more risk averse. For the capital-market setting, we extend the benchmark by introducing seller-side ambiguity about the mortality distribution and robust max-min valuation. Even moderate ambiguity can eliminate the market for static swaps, while dynamic designs remain viable. We then extend the analysis to index-based swaps with basis risk: relative to indemnity swaps, optimal loadings are lower and gains are smaller for both parties, though the static-dynamic preference pattern is unchanged.
In this article, we investigate the optimal insurance demand for an individual under distorted probabilities, considering the participation of government interventions, such as premium subsidies and disaster relief. We model the premium subsidy as a non-decreasing function ranging from 0 to 1, representing the percentage of government support, whereas the relief assistance is characterized by a 1-Lipschitz relief scheme function, reflecting the government’s effort in post-disaster recovery. When the expected-value premium principle is employed, the general form of the optimal retained loss function for the policyholder is derived under a concave government relief scheme. We demonstrate that the optimal retained loss function takes a layered form, shaped by the trade-off between government premium subsidies and relief assistance, and can be further characterized by an ordinary integro-differential equation. In particular, explicit solutions are obtained for VaR and general convex distortion risk measures. To provide further insights, we explore two extensions: one investigates the design of the optimal safety loading from the insurer’s perspective, while the other examines the impact of the government’s budget constraint. Finally, we present numerical examples to illustrate and validate the main findings of the paper.
In practice, insurance companies usually can only offer a small number of policy options in the menu due to cost and management considerations. This paper sheds light on the design of optimal insurance contracts when the number of provided policies is fewer than the types of buyers. Under a monopoly insurance market with complete information, we propose a satisficing mechanism for designing a menu of pooling proportional insurance contracts when the buyers adopt a general risk measure, with the aim of maximizing the seller’s profit while charging premiums by the expected-value premium principle. We introduce an iterative procedure to find these pooling policies, in which only one new policy is introduced in each step. The analytic expressions of the solutions and the associated profits can be effectively derived through the iterative procedure. Numerical examples are presented to illustrate the high efficiency and operational simplicity of the proposed method.
We formulate the optimal reinsurance problem maximising cumulative dividend payments in discrete-time, where our decision is the ceding loss function for each stage, given within a general family. For the surplus, reinsurance is applied to the aggregated loss of each stage and the reinsurance premium is given by a distortion risk measure. Considering dividends as part of the decision variable, we maximise our objective under (a) the surplus, and (b) adding a solvency constraint that controls the ruin probability. Thirdly, we solve a last problem, (c) under both constraints when dividends are given by a dividend rule. For (a)-(b), we find multi-layered optimal policies by minimising the expected loss of the insurer for each stage, moreover, (b) offers a dividend rule as a cap of the surplus. However, the policies of (a)-(b) are not practically justified unless the premium is calculated with coherent distortion risk measures, in which case it is optimal to not reinsure. The optimal policy for (c) with the barrier dividend rule can be found by solving a constrained problem for each stage, where the constraint imposes an upper bound to the retained losses. We obtain multi-layered policies, whose layers cannot be calculated as they depend on Lagrangian multipliers. We propose a Linear Programming (LP) to approximate these optimal policies. We show results for the Expected value, Value-at-Risk, Average-Value-at-Risk and Glue Value-at-Risk. The deductibles we estimate show a relationship with the distortion, the barrier, and the income of the insurer.
The recent resurgence of defined benefit (DB) pensions through cash balance (CB) plans has sparked discussions and captured the attention of employers, unions, and benefits experts. However, there remains limited understanding of how unfreezing DB pensions impacts a firm. To address this gap, we explore the cost and benefit tradeoffs associated with pension unfreezing using a model calibrated to market data within a stochastic framework. Our analysis reveals that the extent to which restarting DB pensions affects a firm’s total pension cost depends on factors such as pension surplus magnitude, employee salaries, firm financial stability, and Pension Benefit Guaranty Corporation (PBGC) premiums. In particular, with a high funding surplus and elevated employee salaries, unfreezing a CB plan can yield greater cost efficiency for companies than pension risk transfer, especially in a high interest rate environment. Our findings provide valuable insights for pension sponsors contemplating the reopening of DB pensions through CB plans, which could potentially counteract the current trends in pension de-risking.
This paper introduces a class of quantile-based, interpretable neural network (NN) models for mortality prediction: the Lee-Carter neural network (LCNN) model, the Renshaw-Haberman neural network (RHNN) model, and the simple neural network (simpleNN) model that balances simplicity with predictive performance. These models preserve the linear interpretability of classic stochastic mortality models while harnessing the flexibility of NNs to capture complex nonlinear patterns in mortality data, thereby achieving enhanced predictive performance. Leveraging a composite loss function that integrates pinball, median anchoring, and quantile-crossing penalty terms, we estimate mortality quantiles and develop an efficient simulation scheme based on interpolation. This framework provides distributional insights essential for pricing, reserving and risk management. Extensive empirical analyses across multiple populations demonstrate that the proposed models consistently outperform traditional approaches in predictive accuracy. We further illustrate their practical utility through an application to longevity swap pricing.
The Accelerated Deaths Model (ADM) builds on the hypothesis that, within a given age cohort, those who are less healthy are more likely to die if infected with Covid-19 than healthier people, leaving a pool of on-average healthier survivors. We use the term 'detrimental selection' which has two complementary aspects: the years of life lost by those who experienced an accelerated death; and the higher average life expectancy of survivors which we call their 'adjusted post-pandemic life expectancy' (ADM's APPLE). Our model represents a novel synthesis of recent advances in our comprehension of mortality heterogeneity and the development of the Proportionality Hypothesis - both of which have improved our understanding of the Covid-19 pandemic. In particular, we identify an important positive relationship between mortality heterogeneity and accelerated deaths. We find, in the case of the Covid-19 pandemic in England, that the years of life lost by those who experienced an accelerated death, while significantly lower than pre-Covid life expectancy, was greater than reported in the media at the time. We also find that the increase in the mean life expectancy of survivors was very small. As a result, the impact on annuity providers (e.g., in terms of potentially higher annuity prices), pension schemes and life insurers was also very small. In contrast, we find that the impact on life expectancy of a general change in future mortality assumptions post-pandemic (i.e., the base mortality table and improvement rate) would be much greater. The ADM has potentially wide application, e.g., to other types of contagion and to climate-related deaths, where we would expect there to be a positive correlation between deaths and all-cause mortality (consistent with the Proportionality Hypothesis), but where the degree of detrimental selection might be different.
Expert knowledge from many different disciplines has the potential to inform on developments that could significantly increase or decrease human life expectancy. However, such knowledge is typically not considered in longevity risk management, since stochastic mortality models are generally only calibrated to historical mortality patterns, i.e., fully data-driven.Following an interdisciplinary approach, we develop a methodology how expert knowledge on the (uncertainty of the) future of human life expectancy can be integrated into the calibration of stochastic mortality models. We argue that this approach is particularly relevant if there are “low probability / high impact” scenarios on the horizon, that are considered plausible by experts in their respective fields but are “virtually impossible” in models calibrated to historical data. Based on current research on treatments that might be effective in slowing down ageing, we motivate and propose an exemplary plausible scenario for the future development of human life expectancy. We assign a potential impact on life expectancy as well as a plausible probability of occurrence to the scenario and present a method for calibrating stochastic mortality models so that the resulting projections are in line with these parameters. In a case study, we analyse and compare the longevity risk in an exemplary annuity portfolio and show that this so-called “driver-driven” calibration can lead to a structurally different assessment of longevity risk than the traditional “data-driven” approach, especially with regard to tail risks.
In the context of insurance risk management, large fluctuations in the surplus process represent a critical source of risk, with implications for the financial stability and resilience of insurers. Understanding and quantifying such variability is therefore essential for assessing the financial robustness of insurers. In this paper, we investigate the range of a L & eacute;vy risk process, which captures surplus variability by tracking the difference between the running supremum and infimum within a given time horizon. We derive new fluctuation identities for the inverse range time under both continuous and Poissonian observation schemes, extending results that were previously available only for Brownian motion in the existing literature. In addition, we study the joint Laplace transform of the inverse range time and the previous extremum time. For illustration, explicit expressions are obtained for the Brownian risk process and the Cram & eacute;r-Lundberg risk model with exponential claims. Finally, we apply our results to the fair valuation of insurance contracts associated with the range process.
In a portfolio of loans, default and prepayment are two competing events, and only the time and type of the first event to occur can be observed. Modeling the competing risks is crucial for mortgage insurance. This paper focuses on modeling the joint distribution of the time to default and the time to prepayment by considering two components: subdistributions of the time to default and time to prepayment, and a copula to model their dependence structure, where the subdistributions are estimated from the portfolio data, and the copula is chosen by concentrating on some optimal criteria.For this purpose, we discuss the compatibility of a copula with the given subdistributions, and provide a method for deriving the marginal distributions of default and prepayment from the subdistributions and a compatible copula. Moreover, two criteria are proposed for finding a copula compatible with the given subdistributions. For estimating the subdistributions, a bilinear model is proposed. The asymptotic properties of the model’s estimators are proved. Additionally, a simulation study demonstrates the consistency of the estimators by considering both large and small sample cases. Finally, an empirical study is performed to estimate the subdistributions with static and time-varying covariates, and to identify compatible copulas under the proposed criteria. The application of the proposed method is further highlighted for determining premiums of mortgage insurance.
We consider bias-corrected estimation of the extreme value index of Pareto-type loss distributions in the censoring framework. The initial estimator is based on a Kaplan-Meier integral from which we remove the bias under a second-order framework. This estimator depends on a suitable external estimation of second-order parameters, which is also discussed. The weak convergence of the bias-corrected estimator is established. It has the nice property of having the same asymptotic variance as the initial estimator. This feature is illustrated in a simulation study where our estimator is compared to alternatives already introduced in the literature. Finally, our methodology is applied to a French non-life insurance dataset.
In this paper, we examine optimal annuitization and asset allocation strategies for a utility-maximizing retiree with constant absolute risk aversion (CARA). The retiree can invest in a market consisting of one risky asset and one risk-free asset and is also allowed to purchase life annuities, with each purchase of life annuities incurring a fixed transaction cost. By using a stochastic control approach and duality techniques, we find that the optimal annuitization strategy is a barrier strategy involving a lower and an upper barrier on the retiree’s wealth. Once the wealth reaches the upper barrier, the retiree purchases additional annuity income to reduce the wealth to the lower one. Furthermore, we provide several numerical examples to illustrate our results and analyze the sensitivity of various parameters. We also compare these optimal strategies with those in the constrained Merton model and the scenario without transaction costs. Numerical results indicate that transaction costs cannot only postpone the retiree’s decision on annuitizing additional wealth, but also result in underspending and slow drawdown rate in the decumulation phases, which provides further explanations for the annuity puzzle, retirement-consumption puzzle and retirement-savings puzzle. Finally, we conduct perturbation analysis and find an asymptotic approximation of the value function when the transaction fee is small.
Ambiguity poses a key challenge in the pricing of catastrophe insurance, as it leads to higher premiums compared to unambiguous risks. This paper proposes a novel approach that integrates model averaging (MA) techniques with an extended a-maxmin framework to address ambiguity in insurance pricing decisions. Specifically, we introduce three MA weighting strategies within a quantile regression setting to mitigate estimation uncertainty and extend the a-maxmin framework to formally incorporate both the insurer's ambiguity aversion and survival constraints into the pricing process. Using earthquake loss data from China (1974-2023), we show that MA improves predictive accuracy and mitigates affordability issues by reducing ambiguity-induced premium inflation, with jackknife model averaging lowering net premiums by 15.69%. Sensitivity analyzes indicate that stronger ambiguity aversion (higher a), tighter survival constraints (lower B), and a higher cost of capital (higher S) all necessitate larger capital reserves to counter bankruptcy risk, thereby raising premiums, with the first two factors exerting a more pronounced influence. The paper offers a coherent toolkit for integrating model uncertainty into catastrophe insurance pricing with practical relevance for risk management and regulation.
As the understanding of GMxB-related risks deepens, insurance companies are increasingly seeking efficient annuity risk management systems. This paper is the first to extend the Karhunen-Lo & egrave;ve (KL) expansion method to the pricing and Greeks estimation of the GMxB variable annuities written on multiple sub-account funds, under the multivariate Ornstein-Uhlenbeck stochastic volatility model. Additionally, the simulation-based pathwise (PW) and likelihood ratio (LR) methods are generalized for efficient Greeks computation within the multi-asset annuity framework. Through asymptotic analysis, we address a theoretical gap in the original KL expansion sampling framework. Numerical experiments demonstrate that the proposed method achieves computational efficiency and robustness, providing a practical and reliable framework for the risk management of complex multi-asset variable annuities.
