Stochastic Mortality Models Research Articles

Irreversible reinsurance: A singular control approach

Reinsurance demand is often investigated using the classical stochastic control framework with the reinsurance level as a regular control variable. However, reinsurance contracts are not traded, and dynamic adjustment of the reinsurance strategy is impractically infeasible. Moreover, reinsurance contracts are long-term commitments and are considered irreversible as they are expensive to close before maturity. We propose a novel irreversible reinsurance framework, where the insurer enters into a sequence of reinsurance contracts at the most appropriate times and the contracts are never reversed afterwards. In each contract, the insurer specifies the amount of reinsurance. We model the insurer's risk exposure using a mean-reverting process as inspired by stochastic mortality modeling and the property loss data from US catastrophes, and then formulate the reinsurance decision as a two-dimensional degenerate singular control problem. Our mathematical problem is closely related to the well-known finite-fuel problem due to the boundedness of the insurer's risk retention. The optimal reinsurance purchase rule is triggered by a free boundary that characterizes the optimal relationship between risk exposure and reinsurance demand. We conduct a numerical analysis to investigate the features of the singular reinsurance strategy and its dependency on the model parameters.

Collective longevity swap: A novel longevity risk transfer solution and its economic pricing

To mitigate the hedger’s longevity risk exposure, this paper proposes a collective longevity swap between a reinsurer (hedge provider) and a group of hedgers (pension plans and annuity providers), and an economic framework to price longevity risk and longevity swaps. Combining the appealing features of two widely discussed longevity swaps, i.e., the indemnity swap and index-based swap, the collective longevity swap requires that reinsurer’s payments are based on the longevity risk of the aggregate portfolio of all hedgers, whereas each hedger could receive indemnity payments on their own portfolio. In a general principal-agent pricing model under the incomplete market setting, we determine the optimal risk premiums and hedge rates that maximize the reinsurer’s expected profit under a set of the hedgers’ participation constraints. We find that, in a competitive market, the proposed collective swap could lead to a higher expected profit for the reinsurer than the indemnity and the index-based swap, and simultaneously improve the benefits of the majority of hedgers without worsening the benefits of the rest. Finally, using a stochastic multi-population mortality model and real-world mortality data, a series of numerical analyses are performed to verify the theoretical findings.

Evaluation of the forecasting accuracy of stochastic mortality models: An analysis of developed and developing countries

This paper evaluates the accuracy performance of eight stochastic mortality models in the forecasting of the male mortality rates pertaining to different age groups and countries. The mortality datasets for three developed countries (Canada, France and Japan) and two developing countries (Taiwan and Ukraine) are employed in this study. For each country, the age range is split into three age groups – A (0–19), B (20–60) and C (61–90). The forecasting accuracy of the mortality models is evaluated using the RMSE, MAE, MPE and MAPE metrics. Mortality models with more complex specifications perform better for the age groups B and C, than for the age group A. The cohort feature is more significant for age categories B and C, especially for the developed countries where there are significant medical and health advances. From an overall perspective, the Lee-Carter, Renshaw-Haberman and Age-Period-Cohort models are superior for the age group A while the Plat model proves to be the best forecasting model for the age categories B and C. The empirical analysis concludes that the mortality patterns diverge for different age categories and countries with different development status. The occurrence of extreme mortality events also negatively affects the patterns of human mortality.

Extensions on the Hatzopoulos–Sagianou Multiple-Components Stochastic Mortality Model

In this paper, we present extensions of the Hatzopoulos–Sagianou (2020) (HS) multiple-component stochastic mortality model. Our aim is to thoroughly evaluate and stress test the HS model by deploying various link functions, using generalised linear models, and diverse distributions in the model’s estimation method. In this work, we differentiate the HS approach by modelling the number of deaths using the Binomial model commonly employed in the literature of mortality modelling. Given this, new HS extensions are derived using the off-the-shelf link functions, namely the complementary log–log, logit and probit, while we also reform the model by introducing a new form of link functions with a particular focus on the use of heavy-tailed distributions. The above-mentioned enhancements conclude to a new methodology for the HS model, and we prove that it is more suitable than those used in the literature to model the mortality dynamics. In this regard, our work offers an extensive experimental testbed to scrutinise the efficiency, explainability and capacity of the HS model extensions using both the off-the-shelf and the newly introduced form of link functions over datasets with different characteristics. The introduced HS extensions bring an improvement by approximately 16% to the model’s goodness-of-fit, while they uncover more fine-grained age clusters. In addition, we compare the performance of the HS extensions against other well-known mortality models, both under fitting and forecast modes. The results reflect the advantageous features of the HS extensions to deliver a highly informative structure and enable the attribution of an identified mortality trend to a unique age cluster. The above-mentioned improvements enable mortality analysts to perform an in-depth and more detailed investigation of mortality trends for specific age clusters and can contribute to the attempts of academia and industry to tackle the uncertainties and risks introduced by the increasing life expectancy.

Application of Continuous Non-Gaussian Mortality Models with Markov Switchings to Forecast Mortality Rates

The ongoing pandemic has resulted in the development of models dealing with the rate of virus spread and the modelling of mortality rates μx,t. A new method of modelling the mortality rates μx,t with different time intervals of higher and lower dispersion has been proposed. The modelling was based on the Milevski–Promislov class of stochastic mortality models with Markov switches, in which excitations are modelled by second-order polynomials of results from a linear non-Gaussian filter. In contrast to literature models where switches are deterministic, the Markov switches are proposed in this approach, which seems to be a new idea. The obtained results confirm that in the time intervals with a higher dispersion of μx,t, the proposed method approximates the empirical data more accurately than the commonly used the Lee–Carter model.

TREE-BASED MACHINE LEARNING METHODS FOR MODELING AND FORECASTING MORTALITY

AbstractMachine learning has recently entered the mortality literature in order to improve the forecasts of stochastic mortality models. This paper proposes to use two pure, tree-based machine learning models: random forests and gradient boosting, based on the differenced log-mortality rates to produce more accurate mortality forecasts. These forecasts are compared with forecasts from traditional, stochastic mortality models and with forecasts from random forests and gradient boosting variants of the stochastic models. The comparisons are based on the Model Confidence Set procedure. The results show that the pure, tree-based models significantly outperform all other models in the majority of cases considered. To address the lack of interpretability issue associated with machine learning models, we demonstrate how to extract information about the relationships uncovered by the tree-based models. For this purpose, we consider variable importance, partial dependence plots, and variable split conditions. Results from the in-sample fit suggest that tree-based models can be very useful tools for detecting patterns within and between variables that are not commonly identifiable with traditional methods.

A Neural Approach to Improve the Lee-Carter Mortality Density Forecasts

Several countries worldwide are experiencing a continuous increase in life expectancy, extending the challenges of life actuaries and demographers in forecasting mortality. Although several stochastic mortality models have been proposed in the literature, mortality forecasting research remains a crucial task. Recently, various research works have encouraged the use of deep learning models to extrapolate suitable patterns within mortality data. Such learning models allow achieving accurate point predictions, though uncertainty measures are also necessary to support both model estimate reliability and risk evaluation. As a new advance in mortality forecasting, we formalize the deep neural network integration within the Lee-Carter framework, as a first bridge between the deep learning and the mortality density forecasts. We test our model proposal in a numerical application considering three representative countries worldwide and for both genders, scrutinizing two different fitting periods. Exploiting the meaning of both biological reasonableness and plausibility of forecasts, as well as performance metrics, our findings confirm the suitability of deep learning models to improve the predictive capacity of the Lee-Carter model, providing more reliable mortality boundaries in the long run.

Breast Cancer Mortality Trends and Predictions to 2030 and Its Attributable Risk Factors in East and South Asian Countries.

BackgroundAmidst the rising breast cancer burden in Asia, we aim to predict the future mortality risk due to breast cancer and identify the risk-attributable deaths for breast cancer among East and South Asian countries.MethodsWe used country-level data to predict the trends in the next decade relating to female breast cancer mortality by employing data from 1990 to 2019 from the Global Burden of Disease 2019 study. We used the stochastic mortality modeling and prediction techniques to forecast the age-specific and risk-attributable breast cancer mortality trends at the regional and national levels of East and South Asia.ResultsThe number of deaths caused by the breast cancer is predicted to increase in East and South Asian countries in the next decade (2020–2030). Age-standardized death rate (ASDR) of breast cancer is predicted to increase by 7.0% from 9.20/100,000 (95% CI: 6.04–12.12) in 1990 to 9.88/100,000 (95% CI: 7.12–11.4) in 2030 in East Asia, and about 35% increase from 13.4/100,000 (95% CI: 9.21–16.02) in 1990 to 18.1/100,000 (95% CI: 13.23–21.10) in 2030 in South Asia. At the national level, the highest percent change in ASDR between 1990 and 2030 was reported in Pakistan (a 62% increase) and Nepal (a 47% increase). The highest percent change in breast cancer mortality between 2020 and 2030 for females of age group 80–84 years was observed in Pakistan [21.6, (95% CI, 20.6–94.7)], followed by Afghanistan [13.3 (4.0–80.8)], and Nepal [36.6 (11.1–125.7)] as compared to the other countries. In the females of aged 50–80 years, the predicted death rates were associated with high body mass index, high-fasting plasma glucose, and diet high in red meat, across the majority of countries under study. Furthermore, reductions in percent change in mortality rates occurred in several countries with increases in sociodemographic index (SDI), notably across high SDI countries.ConclusionBreast cancer mortality risk varies substantially across East and South Asian countries with higher mortality risk in low/middle SDI countries. Early detection using screening, awareness among females and health workers, and cost-effective and timely treatment of patients with breast cancer is vital in stemming the tide of breast cancer in the next decade.

Modeling the Risk in Mortality Projections

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.

Assessing the Impact of the COVID-19 Shock on a Stochastic Multi-Population Mortality Model

We aim to assess the impact of a pandemic data point on the calibration of a stochastic multi-population mortality projection model and its resulting projections for future mortality rates. Throughout the paper, we put focus on the Li and Lee mortality model, which has become a standard for projecting mortality in Belgium and the Netherlands. We calibrate this mortality model on annual death counts and exposures at the level of individual ages. This type of mortality data are typically collected, produced and reported with a significant delay of—for some countries—several years on a platform such as the Human Mortality Database. To enable a timely evaluation of the impact of a pandemic data point, we have to rely on other data sources (e.g., the Short-Term Mortality Fluctuations Data series) that swiftly publish weekly mortality data collected in age buckets. To be compliant with the design and calibration strategy of the Li and Lee model, we transform the weekly mortality data collected in age buckets to yearly, age-specific observations. Therefore, our paper constructs a protocol to ungroup the death counts and exposures registered in age buckets to individual ages. To evaluate the impact of a pandemic shock, like COVID-19 in the year 2020, we weigh this data point in either the calibration or projection step. Obviously, the more weight we place on this data point, the more impact we observe on future estimated mortality rates and life expectancies. Our paper allows for quantifying this impact and provides actuaries and actuarial associations with a framework to generate scenarios of future mortality under various assessments of the pandemic data point.

THE SAINT MODEL: A DECADE LATER

AbstractWhile many of the prevalent stochastic mortality models provide adequate short- to medium-term forecasts, only few provide biologically plausible descriptions of mortality on longer horizons and are sufficiently stable to be of practical use in smaller populations. Among the very first to address the issue of modelling adult mortality in small populations was the SAINT model, which has been used for pricing, reserving and longevity risk management by the Danish Labour Market Supplementary Pension Fund (ATP) for more than a decade. The lessons learned have broadened our understanding of desirable model properties from the practitioner’s point of view and have led to a revision of model components to address accuracy, stability, flexibility, explainability and credibility concerns. This paper serves as an update to the original version published 10 years ago and presents the SAINT model with its modifications and the rationale behind them. The main improvement is the generalization of frailty models from deterministic structures to a flexible class of stochastic models. We show by example how the SAINT framework is used for modelling mortality at ATP and make comparisons to the Lee-Carter model.

Моделювання функцій життєдіяльності та смертності поданих для населення України

In the work the approach to modeling of data sets of the life table is given. Life expectancy limits based on stochastic mortality modeling and the application of the critically low first achievement theory are also investigated. Particular attention is paid to the representation of the function of health, together with a well-established theory of the Force of Mortality, as well as life tables. The parameters of the model are estimated and analyzed according to the data of demographic tables for the population of Ukraine.

Valuation of a Guaranteed Minimum Maturity Benefit in a Self-exciting Stochastic Mortality Model

Valuation of a Guaranteed Minimum Maturity Benefit in a Self-exciting Stochastic Mortality Model

Longevity risk analysis: applications to the Italian regional data

<abstract><p>Longevity risk is the risk that members of a given population will live longer than expected. When it occurs, pension providers may have to pay pensions for longer than expected, significantly increasing their costs. While this risk is being adequately studied using the national mortality data provided by the Human Mortality Database, relatively few studies exist that analyse sub-national data. This manuscript proposes a comparative study of some stochastic mortality models to measure the longevity risk on Italian mortality data at the regional level. In particular, the use of the Lee-Carter and Li-Lee models is explored. The models are compared in fitting quality, forecasting accuracy and complexity. Numerical experiments and applications to immediate life annuity evaluation are presented.</p></abstract>

Linking Pensions to Life Expectancy: Tackling Conceptual Uncertainty through Bayesian Model Averaging

Linking pensions to longevity developments at retirement age has been one of the most common policy responses of pension schemes to aging populations. The introduction of automatic stabilizers is primarily motivated by cost containment objectives, but there are other dimensions of welfare restructuring in the politics of pension reforms, including recalibration, rationalization, and blame avoidance for unpopular policies that involve retrenchments. This paper examines the policy designs and implications of linking entry pensions to life expectancy developments through sustainability factors or life expectancy coefficients in Finland, Portugal, and Spain. To address conceptual and specification uncertainty in policymaking, we propose and apply a Bayesian model averaging approach to stochastic mortality modeling and life expectancy computation. The results show that: (i) sustainability factors will generate substantial pension entitlement reductions in the three countries analyzed; (ii) the magnitude of the pension losses depends on the factor design; (iii) to offset pension cuts and safeguard pension adequacy, individuals will have to prolong their working lives significantly; (iv) factor designs considering cohort longevity markers would have generated higher pension cuts in countries with increasing life expectancy gap.

POINT AND INTERVAL FORECASTS OF DEATH RATES USING NEURAL NETWORKS

AbstractThe Lee–Carter model has become a benchmark in stochastic mortality modeling. However, its forecasting performance can be significantly improved upon by modern machine learning techniques. We propose a convolutional neural network (NN) architecture for mortality rate forecasting, empirically compare this model as well as other NN models to the Lee–Carter model and find that lower forecast errors are achievable for many countries in the Human Mortality Database. We provide details on the errors and forecasts of our model to make it more understandable and, thus, more trustworthy. As NN by default only yield point estimates, previous works applying them to mortality modeling have not investigated prediction uncertainty. We address this gap in the literature by implementing a bootstrapping-based technique and demonstrate that it yields highly reliable prediction intervals for our NN model.

A Square-Root Factor-Based Multi-Population Extension of the Mortality Laws

In this paper, we present and calibrate a multi-population stochastic mortality model based on latent square-root affine factors of the Cox-Ingersoll and Ross type. The model considers a generalization of the traditional actuarial mortality laws to a stochastic, multi-population and time-varying setting. We calibrate the model to fit the mortality dynamics of UK males and females over the last 50 years. We estimate the optimal states and model parameters using quasi-maximum likelihood techniques.

Lee-Carter method for forecasting mortality for Peruvian Population

In this article, we have modeled mortality rates of Peruvian female and male populations during the period of 1950-2017 using the two-factor Lee-Carter (LC) model. The stochastic mortality model was introduced by Lee and Carter (1992) and has been used by many authors for fitting and forecasting the human mortality rates. The Singular Value Decomposition (SVD) approach is used for estimation of the parameters of the LC model. Utilizing the best fitted auto regressive integrated moving average (ARIMA) model we forecast the values of the time dependent parameter of the LC model for the next thirty years. The forecasted values of life expectancy at different age group with 95% confidence intervals are also reported for the next thirty  years. In this research we use the data, obtained from the Peruvian National Institute of Statistics (INEI). &nbsp

Optimal dynamic longevity hedge with basis risk

This paper proposes an optimal dynamic strategy for hedging longevity risk in a discrete-time setting. Our proposed hedging strategy relies on standardized mortality-linked securities and minimizes the variance of the hedging error as induced by the population basis risk. While the formulation of our proposed hedging strategy is quite general, we use a stylized pension plan, together with a specified “yearly rollingǥ trading strategy involving q-forwards and a specified stochastic mortality model, to illustrate our proposed strategy. Under these specifications, we show that the resulting hedging problem can be formulated as a stochastic optimal control framework and that a semi-analytic solution can be derived through an extended Bellman equation. Extensive Monte Carlo studies are conducted to highlight the effectiveness of our proposed hedging strategy. We also consider a scheme to approximate the semi-analytic solution in order to reduce the computational time significantly while still retaining its hedge effectiveness. We benchmark our strategy against the “delta” hedging strategy as well as its robustness to q-forwards maturity, reference age, interest rate, and stochastic mortality models. The proposed strategy has many appealing features, including its discrete-time setting which is consistent with market practice and hence conducive to practical implementation, and its generality in that the underlying hedging principle can be applied to other standardized mortality-linked securities and other stochastic models.

Tail index-linked annuity: A longevity risk sharing retirement plan

This paper proposes an innovative retirement product focusing on longevity risk sharing, a contract we refer to as tail index-linked annuity (TILA). Specifically, the proposed TILA pays out variable annual payments, which will be equal to a regular nominal amount when a reference survival index is lower than a predetermined threshold (i.e. normal evolution of longevity risk), and a reduced, index-dependent payment when the threshold is passed (i.e. highly unfavorable evolution of longevity risk). The proposed TILA aims at not only improving the benefits of the policyholders, which has been the focus in recent literature on innovative retirement products, but also reducing the longevity risk exposure of the insurer, particularly for advanced retirement ages. Using real-world mortality data and a stochastic multi-population mortality model, we find that the proposed TILA leads to higher expected lifetime utility than regular annuities for policyholders with different degrees of risk aversions. Meanwhile, numerical analysis shows that the proposed TILA could greatly mitigate the solvency risk of the insurer, leading to a substantially lower loss probability and expected (tail-) loss than regular annuities in the presence of a longevity shock, and therefore could reduce the insurer's required solvency capital under the latest solvency regulations.