Frailty-based mortality models and reserving for longevity risk

Human lifespan increments represent one of the main current risks for governments and pension and health benefits providers. Longevity societies imply financial sustainability challenges to guarantee adequate socioeconomic conditions for all individuals for a longer period. Consequently, modelling population dynamics and projecting future longevity scenarios are vital tasks for policymakers. As an answer, the demographic and the actuarial literature have been introduced and compared to several stochastic mortality models, although few studies have thoroughly tested the uncertainty concerning mortality projections. Forecasting mortality uncertainty levels have a central role since they reveal the potential, unexpected longevity rise and the related economic impact. Therefore, the present study poses a methodological framework to backtest uncertainty in mortality projections by exploiting uncertainty metrics not yet adopted in mortality literature. Using the data from the Human Mortality Database of the male and female populations of five countries, we present some numerical applications to illustrate how the proposed criterion works. The results show that there is no mortality model overperforming the others in all cases, and the best model choice depends on the data considered.

In this paper we mirror the framework of generalized (non-)linear models to define the family of generalized age-period-cohort stochastic mortality models which encompasses the vast majority of stochastic mortality projection models proposed to date, including the well-known Lee-Carter and Cairns-Blake-Dowd models. We also introduce the R package StMoMo which exploits the unifying framework of the generalized age-period-cohort family to provide tools for fitting stochastic mortality models, assessing their goodness of fit and performing mortality projections. We illustrate some of the capabilities of the package by performing a comparison of several stochastic mortality models applied to the England and Wales population.

Capturing the Uncertainty in Long-Term Mortality Forecasts The uncertainty in future longevity presents a substantial risk factor for insurance companies, pension funds, and retirement systems. In “Modeling the Risk in Mortality Projections,” Zhu and Bauer present novel stochastic models for analyzing this longevity risk that focus on the uncertainty associated with long-term mortality projections and capture the evolution of mortality forecasts over the past decades. They arrive at their models by analyzing time series of mortality forecasts in a forward modeling framework, which contrasts with conventional stochastic mortality models that are built on age-specific realized mortality rates. The authors showcase their models in exemplifying financial applications in both traditional life insurance markets and the emerging longevity risk transfer market. A key takeaway is that uncertainty in positions that depend on the long-term evolution of mortality is substantially greater under their models than suggested by conventional models.

Chapter 4: ‘Helping Savers to Manage Longevity Risk’

Pension schemes all over the world are under increasing pressure to efficiently hedge longevity risk imposed by ageing populations. In this work, we study an optimal investment problem for a defined contribution pension scheme that decides to hedge longevity risk using a mortality-linked security, typically a longevity bond. The pension scheme promises a minimum guarantee which allows the members to purchase lifetime annuities upon retirement. The scheme manager invests in the risky and riskless assets available on the market, including the longevity bond. We transform the corresponding constrained optimal investment problem into a single investment portfolio optimization problem by replicating future contributions from members and the minimum guarantee provided by the scheme. We solve the resulting optimization problem using the dynamic programming principle. Through a series of numerical studies, we show that the longevity risk has an important impact on the investment strategy performance. Our results add to the growing evidence supporting the use of mortality-linked securities for efficient hedging of longevity risk.

This study sets out a backtesting framework applicable to the multiperiod-ahead forecasts from stochastic mortality models and uses it to evaluate the forecasting performance of six different stochastic mortality models applied to English & Welsh male mortality data. The models considered are the following: Lee-Carter’s 1992 one-factor model; a version of Renshaw-Haberman’s 2006 extension of the Lee-Carter model to allow for a cohort effect; the age-period-cohort model, which is a simplified version of Renshaw-Haberman; Cairns, Blake, and Dowd’s 2006 two-factor model; and two generalized versions of the last named with an added cohort effect. For the data set used herein, the results from applying this methodology suggest that the models perform adequately by most backtests and that prediction intervals that incorporate parameter uncertainty are wider than those that do not. We also find little difference between the performances of five of the models, but the remaining model shows considerable forecast instability.

BackgroundPrecise predictions of incidence and mortality rates due to breast cancer (BC) are required for planning of public health programs as well as for clinical services. A number of approaches has been established for prediction of mortality using stochastic models. The performance of these models intensely depends on different patterns shown by mortality data in different countries.MethodsThe BC mortality data is retrieved from the Global burden of disease (GBD) study 2017 database. This study include BC mortality rates from 1990 to 2017, with ages 20 to 80+ years old women, for different Asian countries. Our study extend the current literature on Asian BC mortality data, on both the number of considered stochastic mortality models and their rigorous evaluation using multivariate Diebold-Marino test and by range of graphical analysis for multiple countries.ResultsStudy findings reveal that stochastic smoothed mortality models based on functional data analysis generally outperform on quadratic structure of BC mortality rates than the other lee-carter models, both in term of goodness of fit and on forecast accuracy. Besides, smoothed lee carter (SLC) model outperform the functional demographic model (FDM) in case of symmetric structure of BC mortality rates, and provides almost comparable results to FDM in within and outside data forecast accuracy for heterogeneous set of BC mortality rates.ConclusionConsidering the SLC model in comparison to the other can be obliging to forecast BC mortality and life expectancy at birth, since it provides even better results in some cases. In the current situation, we can assume that there is no single model, which can truly outperform all the others on every population. Therefore, we also suggest generating BC mortality forecasts using multiple models rather than relying upon any single model.

This study aims to investigate the impact of China’s new fertility policy on the actuarial balance of its Urban Employee Basic Endowment Insurance (UEBEI) fund, with stochastic mortality model included to address the longevity risk. Combined with the latest UEBEI policy, this paper constructs an actuarial balance model and introduces the growth rate of wage, the age of employment and the age of retirement, the rate of payment, the rate of replacement, the annual rate of pension adjustment, and the population in terms of age into the model, which arise from the rate of payment, average wage, and personal and social factors. This study uses the sixth population census data by age group and gender to employ empirical analysis. The sensitivity analysis of China’s basic pension insurance fund balance is made. It is concluded that the increase in the growth rate of wage caused by social factors, the increase of fund investment returns, the delay of retirement, and the increase in the fund collection rate are all conducive to the sustainability of the UEBEI fund.

Purpose The purpose of this paper is to present a clear path to securitize the longevity risk with two distinct swaps in order to inspire a new Chinese life market. Design/methodology/approach Studies on longevity risk securitization consist of three aspects, respectively, instrument design, pricing methodology and mortality projection. The swaps designed are referenced, respectively, to vanilla and complex survivor swaps (Dowd et al., 2006; Lin and Cox, 2005). Methods applied are RHH model and Gompertz law for mortality projection, as well as two-factor Wang transformation for pricing. Findings This paper figures out the market price of risk in Chinese annuity market, checks for the sensitivity of the price to parameters and tests the hedging effects by Monte Carlo simulation. Originality/value Based on the theoretical and numerical results, this paper suggests an effective way to possibly witness the birth of New Life Market in China.

Forecasting longevity is essential in multiple research and policy areas, including the pricing of life insurance contracts, the valuation of capital market solutions for longevity risk management, and pension policy. This paper empirically investigates the predictive performance of Recurrent Neural Networks (RNN) with Gated Recurrent Unit (GRU) and Long Short-Term Memory (LSTM) architectures in jointly modeling and multivariate time series forecasting of age-specific mortality rates at all ages. We fine-tune the three hidden layers GRU and LSTM model’s hyperparameters for time series forecasting and compare the model’s forecasting accuracy with that produced by traditional Generalised Age-Period-Cohort (GAPC) stochastic mortality models. The empirical results suggest that the two RNN architectures generally outperform the GAPC models investigated in both the male and female populations, but the results are sensitive to the accuracy criteria. The empirical results also show that the RNN-GRU network slightly outperforms the RNN with an LSTM architecture and can produce mortality schedules that capture relatively well the dynamics of mortality rates across age and time. Further investigations considering other RNN architectures, calibration procedures, and sample datasets are necessary to confirm the superiority of RNN in forecasting longevity.

ABSTRACTGroup self‐annuitization (GSA) schemes are designed to share uncertain future mortality experience including systematic improvements. Challenges for designing group pooled schemes include decreasing average payments when mortality improves significantly, decreasing numbers in the pool at older ages, and the impact of dependence from systematic mortality improvements across different ages of members in the pool. This article uses a multiple‐factor stochastic mortality model in a simulation study to show how pooling can be made more effective and to quantify the limitations of these pooling schemes arising from the impact of systematic longevity risk.

The question of how to effectively alleviate the financial pressure on pension insurance due to the increase in life expectancy has become an important issue in the reform of China’s social security system. This paper introduced two life expectancy-related delayed retirement schemes, namely the fixed expected retirement residual life and the fixed life burden ratio. We modeled the financial balance of the employee pension fund and the pension wealth of employees with a dynamic retirement age according to pension policy. Using the population mortality data, the dynamic retirement age under the two schemes was estimated under the stochastic mortality model. Following this, the impact of the two delayed retirement schemes was quantitatively assessed from the perspectives of the financial sustainability of the pension fund and the pension wealth of employees using insurance actuarial methods. This study found that the two life expectancy-related delayed retirement schemes have obvious effects on reducing the gap between the income and expenditure of the pension fund and increasing the pension wealth of employees. Moreover, it found that the fixed expected retirement residual life program contributes more than the fixed life burden ratio program to improve the financial sustainability of the pension fund and the pension wealth benefits of employees.

Mortality Density Forecasts: An Analysis of Six Stochastic Mortality Models

This research examines the diversity-productivity relationship in a semiarid scrubland, initially under late successional conditions and subsequently under early successional conditions created by experimental clearing, to explore the roles that productivity and stochastic mortality play in species exclusion in this environment. A total of fifteen plots were studied by measuring environmental conditions and biomass components of shrubs and seedlings. These stands were distributed along a productivity gradient across five different landforms. A hypothesis about the stochastic self-thinning mortality model along the gradient was evaluated with the diversity-productivity-environment data. The diversity-productivity relationship was linear and reversed between the early and late succession stages. The hypothesis of stochastic mortality of species exclusion was rejected in the early stages of succession and partially accepted in the mature stage of succession. Species exclusion was negatively related to productivity gradients, suggesting that strong interspecific competition occurs in high productivity plots and that a larger number of species can survive in higher abiotic stress landscapes. Further research is needed to understand the temporal and spatial variations of the ecological interactions that shape this plant community.

The insurance enterprise faces growing challenges in optimizing annuity pricing strategies under changing market situations and growing opposition to its product portfolios. Conventional actuarial models have massive deficiencies, as they depend upon historical data patterns and linear assumptions that cannot capture complex, nonlinear relationships among risk factors and profitability results. The usage of machine learning algorithms in annuity pricing frameworks truly addresses core shortcomings in conventional methodologies, with particular regard to mortality risk assessment, sensitivity analysis of interest rates, and policyholder behavior prediction. Gradient boosting decision trees, neural network architectures, and ensemble methods applied to fixed and indexed annuity products achieve superior predictive accuracy compared to generalized linear models commonly found in traditional actuarial practice. This implementation framework integrates machine learning predictive capabilities with established actuarial practices to ensure regulatory compliance while maintaining mathematical soundness. Deep learning approaches to mortality forecasting, especially Long Short-Term Memory networks, transcend restrictions of classical Lee-Carter models by allowing temporal dependencies and nonlinear patterns to be captured, characteristic of modern mortality experiences. Reinforcement learning applications to derivative hedging strategies optimize dynamic rebalancing decisions for indexed annuity products with embedded options. Quantitative comparison among stochastic mortality models across heterogeneous populations reveals performance differences conditional upon demographic characteristics and projection horizons. Spread margin enhancement through predictive analytics allows a more sophisticated crediting rate determination and policyholder retention strategy. ROI considerations include not only direct improvements in profitability but also indirect operational efficiency gains, balanced against infrastructure and personnel investment requirements.