Stochastic Mortality Models Research Articles

Stochastic mortality model with respect to mixed fractional Poisson process: Calibration and empirical analysis of long-range dependence in actuarial valuation

Recently, many studies have adopted the fractional stochastic mortality process in characterising the long-range dependence (LRD) feature of mortality dynamics, while there are still fewer appropriate non-Gaussian fractional models to describe it. We propose a stochastic mortality process driven by a mixture of Brownian motion and modified fractional Poisson process to capture the LRD of mortality rates. The survival probability under this new stochastic mortality model keeps flexibility and consistency with existing affine-form mortality models, which makes the model convenient in evaluating mortality-linked products under the market-consistent method. The formula of survival probability also considers the historical information from survival data, which enables the model to capture historical health records of lives. The LRD feature is reflected by our proposed model in the empirical analysis, which includes the calibration and prediction of survival curves based on recent generation data in Japan and the UK. Finally, the consequent empirical analysis of annuity pricing illustrates the difference of whether this feature is involved in actuarial valuation.

Solvency analysis of deferred annuities

AbstractWhile the solvency analysis of immediate life annuity portfolios has been extensively studied, the case of deferred annuities has received comparatively much less attention. We assess the importance and effect of stochastic mortality models and interest rates on the solvency analysis of a portfolio of deferred annuity contracts. Our analysis considers three steps: first, the benchmark case where mortality rates and interest rates are both deterministic; then, the case in which only mortality rates are stochastic is explored; finally, the full model where both mortality rates and interest rates are stochastic. The results demonstrate the model risk stemming from the uncertainty in the mortality models and its impact on the evaluation of solvency margins for life annuities. The role of the deferment period is thoroughly discussed and compared to the case of immediate annuities.

On the benefits of pension plan consolidation: Understanding the impact of full plan mergers

Abstract This study investigates the benefits and drawbacks of pension plan consolidation by quantifying the impact of mergers of heterogeneous plans on different stakeholders in a unique Canadian implementation of defined benefit plans. Using a comprehensive framework that combines a realistic economic scenario generator, a stochastic mortality model that captures differences among subpopulations, a cost model with economies of scale, and a dynamic asset allocation methodology, we evaluate the combined effect of asset- and liability-side changes on three groups of measures: plan-related risk measures assessing profits from an economic capital perspective, consumption-based metrics to understand the impact on members, and contribution risk measures capturing the risk from the employer’s viewpoint. We apply the framework to a hypothetical and empirically relevant merger and find that consolidation is favorable under most circumstances: the positive impacts of better diversification and economies of scale continue to outweigh the negative effects of heterogeneity even when the merging plans have different mortality expectations, different maturity levels, or modest differences in initial funded ratios.

Frailty-based mortality models and reserving for longevity risk

For the life insurance industry and pension schemes, mortality projections are critical for accurately managing exposure to longevity risk in terms of both premium setting and reserving. Frailty has been identified as an important latent factor underpinning the evolution of mortality rates. It represents the comorbidities that drive the deterioration of the human body’s physiological capacity. In this paper, we propose a stochastic mortality model that incorporates the trend in frailty, and we analyse the gap between the actuarial evaluations of premiums and technical provisions calculated under frailty-based and traditional stochastic mortality models. We observe that the frailty-based model leads to higher levels of uncertainty in estimates and projections (compared to a traditional stochastic mortality model), which is attributed to the explicit modelling of the comorbidities. This leads to proposing a potentially important policy-oriented recommendation: the incorporation of frailty in mortality modelling would allow for the profiling of mortality according to the portfolio in force for the insurer (or pension scheme), thereby mitigating the problem of adverse selection.

Modeling and forecasting mortality with economic, environmental and lifestyle variables

AbstractTraditional stochastic mortality models tend to extrapolate, to focus on identifying trends in mortality without explaining them. Those that do link mortality with other variables usually limit themselves to GDP. This article presents a novel stochastic mortality model that incorporates a wide range of variables related to economic, environmental and lifestyle factors to predict mortality. The model uses principal components derived from these variables, extending the Niu and Melenberg (Demography 51(5):1755–1773, 2014) model to variables other than GDP, and is applied to 37 countries from the Human Mortality Database. Model fit is superior to the Lee–Carter model for 18 countries. The forecasting accuracy of the proposed model is better than that of the Niu–Melenberg model for half of the countries analyzed under various jump-off years. The model highlights the importance of economic prosperity and healthy lifestyle choices in improving lifespan, while the effect of environmental variables is mixed. By clarifying the specific contributions of different factors and thus making trade-offs explicit, the model is designed to facilitate scenario building and policy planning.

Multi-population mortality modelling and forecasting with divergence bounds

AbstractUnderstanding the mortality dynamics and forecasting its future evolution is crucial for insurance companies and governments facing the risk that individuals might live longer than expected (the so-called longevity risk). This paper introduces a neural network model that allows an accurate modelling and forecasting of the mortality rates of many populations. The neural network model we propose is designed to present a fully explainable structure, allowing for understanding how predictions are formulated. Furthermore, the model addresses the problem of measuring and managing the divergence of the long-term forecasts of the mortality rates arising when one decides to model the mortality of two or more populations simultaneously. Indeed, for many models available in the literature, this divergence grows over time, resulting in an ever-increasing trend in the gap in life expectancy among countries that appear unrealistic and biologically unreasonable. The proposed model allows the construction of analytical bounds for this divergence and illustrates that these bounds can be exploited to analyse and measure the dissimilarities between two or more populations and identify opportunities for longevity risk diversification. Numerical experiments performed using all the data from the Human Mortality Database data show that our model produces more accurate mortality forecasts with respect to some well-known stochastic mortality models and allows us to obtain valuable insights about the mortality pattern of the population considered.

Assessing Delayed Retirement Policies Linked to Dynamic Life Expectancy with Stochastic Dynamic Mortality

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.

Frailty-based Lee–Carter family of stochastic mortality models

AbstractIn the actuarial literature, frailty is defined to be the unobserved variable which encompasses all the factors affecting human mortality other than gender and age. Heterogeneity in individual frailty can play a significant role in population mortality dynamics. In the present paper, we identify the main latent factors that explain the frailty component, in order to clarify its role in mortality projections. We show, using longitudinal survey data, that frailty is mainly due to co-morbidities that impact on the process of deterioration in terms of the human body’s physiological capacity. Accordingly, we provide frailty-based stochastic models for projecting mortality based on the Lee–Carter family of models. We propose several versions that consider frailty both as an age-dependent and a time-dependent factor and also combining the interaction effects of age and time in comparison with the general level of mortality, and compare the resulting mortality projections using data from England.

Multi-Population O’Hare with ARIMA, ARIMA-GARCH and ANN in Forecasting Mortality Rate

Multi-population mortality model has gained attention from prominent researchers of mortality due to its ability to provide biologically reasonable forecast. Previously, many researchers have proposed several multi-population stochastic mortality models that they considered adequate to produce accurate life expectancy. However, little have been addressed of the variability in full ages and time, which can contribute to an erroneous estimation of life expectancy. Therefore, this study proposed a new multi-population O’Hare with ARIMA, ARIMA-GARCH and ANN in forecasting the mortality rate for male and female in Malaysia, Taiwan, Japan, Hong Kong, Australia, USA, UK, Canada, and Switzerland. Multi-population O’Hare was used as a reference model, whilst ARIMA, ARIMA-GARCH and ANN were incorporated to the reference model to forecast the mortality rates. The adequacy of the proposed model was assessed by using measurement errors which were Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE). The results showed by multi-population O’Hare with ARIMA-GARCH gave the best forecasting performance for Taiwan, Japan, Australia, USA, UK, Canada, and Switzerland. On the other hand, multi-population O’Hare with ARIMA gave the best forecasting performance for Malaysia, whereas multi-population O’Hare with ANN gave the best forecasting performance for Hong Kong.

Disentangling Trend Risk and Basis Risk with Functional Time Series

In recent multi-population stochastic mortality models, one critical scientific issue is the vague distinction between trend risk and population basis risk. In particular, the cross- and auto-correlations between the innovations of the latent factors representing the common trend and the population-specific trends are often assumed to be non-existent, although they are possibly statistically significant. While it is theoretically possible to capture such correlations by treating the latent factors as a vector time series, the resulting model would contain a large number of parameters, which may in turn lead to robustness problems. In this paper, we address these issues by the use of the product–ratio model. Contrary to the prevalent assumption of non-existent correlations, the latent factors under the product–ratio model are approximately uncorrelated. This permits us to disentangle trend risk and population basis risk, thereby sparing us from the need to use a heavily parameterized vector time-series process. Compared to the augmented common factor model, our approach demonstrates improved robustness in terms of correlation structures and hedging performance, offering a new perspective on treating cross- and auto-correlations between latent factors in mortality modeling.

Guaranteed Minimum Maturity Benefits in a Self-Exciting Stochastic Mortality Model: Pricing, Estimation and Calibration

The guaranteed minimum maturity benefit (GMMB) is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present article we assume a self-exciting behavior for the mortality dynamics, described by a Hawkes-type process with exponential kernel. This allows us to keep both the Markov and affine features but introduces jumps with a stochastic intensity. These types of dynamics, exhibiting a jump clustering property, are quite convenient for describing mortality in some critical situations, like epidemics, where contagion phenomena make the probability of jump arrivals higher whenever a jump occurs. By assuming diffusion-type dynamics for both the fund and the interest rate and introducing all of the possible correlations among the diffusion processes necessary to obtain realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the value of a GMMB. Lastly, we calibrate our model and perform an empirical analysis to determine the effects of excess mortality on GMMB prices as well as a death benefit portfolio.

Should Selection of the Optimum Stochastic Mortality Model Be Based on the Original or the Logarithmic Scale of the Mortality Rate?

Stochastic mortality models seek to forecast future mortality rates; thus, it is apparent that the objective variable should be the mortality rate expressed in the original scale. However, the performance of stochastic mortality models—in terms, that is, of their goodness-of-fit and prediction accuracy—is often based on the logarithmic scale of the mortality rate. In this article, we examine whether the same forecast outcomes are obtained when the performance of mortality models is assessed based on the original and log scales of the mortality rate. We compare four different stochastic mortality models: the original Lee–Carter model, the Lee–Carter model with (log)normal distribution, the Lee–Carter model with Poisson distribution and the median Lee–Carter model. We show that the preferred model will depend on the scale of the objective variable, the selection criteria measure and the range of ages analysed.

Intergenerational actuarial fairness when longevity increases: Amending the retirement age

Continuous longevity improvements and population ageing have led countries to modify national public pension schemes by increasing standard and early retirement ages in a discretionary, scheduled, or automatic way, and making it harder for people to retire prematurely. To this end, countries have adopted alternative retirement age strategies, but our analyses show that the measures taken are often poorly designed and consequently misaligned with the pension scheme's ultimate goals. This paper discusses how to implement automatic indexation of the retirement age to life expectancy developments while respecting the principles of intergenerational actuarial fairness and neutrality among generations of the respective policy scheme design. With stable demographic conditions, we show in policy designs in which extended working lives translate into additional pension entitlements, the pension age must be automatically updated to keep the period in retirement constant. Alternatively, policy designs that pursue a fixed replacement rate are consistent with retirement age policies targeting a constant balance between active years in the workforce and years in retirement. Under conditions of population ageing, the statutory pension age will have to increase at a faster rate to meet the intergenerational equity criteria. The empirical strategy employed a Bayesian Model Ensemble approach to stochastic mortality modelling to address model risk and generate forecasts of intergenerationally and actuarially fair pension ages for 23 countries from 2000 to 2050. The findings show that the pension age increases needed to accommodate the effect of longevity developments on pay-as-you-go equilibrium and to reinstate equity between generations are sizeable and well beyond those employed and/or legislated in most countries. A new wave of pension reforms may be at the doorsteps.

Hedging longevity risk under non-Gaussian state-space stochastic mortality models: A mean-variance-skewness-kurtosis approach

Longevity risk has recently become a high profile risk among insurers and pension plan sponsors. One way to mitigate longevity risk is to build a hedge using derivatives that are linked to mortality indexes. Longevity hedging methods are often based on the normality assumption, considering only the variance but no other (higher) moments. However, strong empirical evidence suggests that mortality improvement rates are driven by asymmetric and fat-tailed distributions, so that existing longevity hedging methods should be expanded to incorporate higher moments. This paper fills the gap by adopting a mean-variance-skewness-kurtosis approach based on non-Gaussian extensions of commonly-used stochastic mortality models, formulated in a state-space setting. On the basis of a general representation of these models, the authors derive (approximate) analytical expressions for the moments of the present values of the hedging instruments and the liability being hedged. These expressions are then integrated with a polynomial goal programming model, from which the optimal hedge portfolio is identified. Finally, the paper demonstrates the theoretical results with a real mortality data set and a range of hedger preferences.

Enhancing diagnostic of stochastic mortality models leveraging contrast trees: an application on Italian data

The rise in longevity in the twentieth century has led to a growing interest in modeling mortality, and new advanced techniques such as machine learning have recently joined to more traditional models, such as the Lee–Carter or the Age Period Cohort. However, the performances of these models, in terms of fitting to the observed data, are difficult to compare in a unified framework. The goodness-of-fit measures summarizing the discrepancy between the estimates from the model and the observed values are different for traditional mortality models and machine learning. We, therefore, employ a new technique, Contrast trees, which, leveraging on decision trees, provides a general approach for evaluating the quality of fit of different kinds of models by detecting the regions in the input space where models work poorly. Once the low-performance regions are detected, we use Contrast boosting to improve the inaccuracies of mortality estimates provided by each model. To verify the ability of this approach, we consider both standard stochastic mortality models and machine learning algorithms in the estimate of the Italian mortality rates from the Human Mortality Database. The results are discussed using both graphical and numerical tools, with particular attention to the high-error regions.

Some comments on “A Hermite spline approach for modelling population mortality” by Tang, Li & Tickle (2022)

AbstractTang et al. (2022) propose a new class of models for stochastic mortality modelling using Hermite splines. There are four useful features of this class that are worth emphasising. First, for single-sex datasets, this new class of projection models can be fitted as a generalised linear model. Second, these models can automatically extrapolate mortality rates to ages above the maximum age of the data set. Third, simpler sub-variants of the models exist for forecasting when one of the variables lacks a clear drift. Finally, a minor reparameterisation increases the quality of long-range forecasts of period mortality.

A Preliminary Investigation of a Single Shock Impact on Italian Mortality Rates Using STMF Data: A Case Study of COVID-19

Mortality shocks, such as pandemics, threaten the consolidated longevity improvements, confirmed in the last decades for the majority of western countries. Indeed, just before the COVID-19 pandemic, mortality was falling for all ages, with a different behavior according to different ages and countries. It is indubitable that the changes in the population longevity induced by shock events, even transitory ones, affecting demographic projections, have financial implications in public spending as well as in pension plans and life insurance. The Short Term Mortality Fluctuations (STMF) data series, providing data of all-cause mortality fluctuations by week within each calendar year for 38 countries worldwide, offers a powerful tool to timely analyze the effects of the mortality shock caused by the COVID-19 pandemic on Italian mortality rates. This dataset, recently made available as a new component of the Human Mortality Database, is described and techniques for the integration of its data with the historical mortality time series are proposed. Then, to forecast mortality rates, the well-known stochastic mortality model proposed by Lee and Carter in 1992 is first considered, to be consistent with the internal processing of the Human Mortality Database, where exposures are estimated by the Lee–Carter model; empirical results are discussed both on the estimation of the model coefficients and on the forecast of the mortality rates. In detail, we show how the integration of the yearly aggregated STMF data in the HMD database allows the Lee–Carter model to capture the complex evolution of the Italian mortality rates, including the higher lethality for males and older people, in the years that follow a large shock event such as the COVID-19 pandemic. Finally, we discuss some key points concerning the improvement of existing models to take into account mortality shocks and evaluate their impact on future mortality dynamics.

Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa

Longevity is without a doubt on the rise throughout the world due to advances in technology and health. Since 1960, Ghana’s average annual mortality improvement has been about 1.236%. This poses serious longevity risks to numerous longevity-bearing assets and liabilities. As a result, this research investigates the effect of mortality improvement on pension annuities related to a particular pension scheme in Ghana. Different stochastic mortality models (Lee–Carter, Renshaw–Haberman, Cairns–Blake–Dowd, and Quadratic Cairns–Blake–Dowd) are used to forecast mortality improvements between 2021 and 2030. The results from accuracy metrics indicate that the quadratic Cairns–Blake–Dowd model exhibits the best fit to the mortality data. The findings from the study demonstrate that mortality for increasing ages within the retirement period was declining, with increasing improvement associated with increasing ages. Furthermore, the forecasts were used to estimate the associated single benefit annuity for a GHS 1 per annum payment to pensioners, and it was discovered that the annuity value expected to be paid to such people was not significantly different regardless of the pensioner’s current age.

A Bibliometric Analysis of Research on Stochastic Mortality Modelling and Forecasting

Mortality improvements and life expectancies have been increasing in recent decades, leading to growing interest in understanding mortality risk and longevity risk. Studies of mortality forecasting are of interest among actuaries and demographers because mortality forecasting can quantify mortality and longevity risks. There is an abundance of literature on the topic of modelling and forecasting mortality, which often leads to confusion in determining a particular model to be adopted as a reliable tool. In this study, we conducted a bibliometric analysis with a focus on citation and co-citation analyses and co-occurrences of keywords to determine the most widely used stochastic mortality model. We found that the Lee–Carter model has remained one of the most relevant mortality models since its development in the 1990s. Furthermore, we also aimed to identify emerging topics and trends relating to mortality modelling and forecasting based on an analysis of authors’ keywords. This study contributes to the literature by providing a comprehensive overview and evolution of publications in stochastic mortality modelling and forecasting. Researchers can benefit from the present work in determining and exploring emerging trends and topics for future studies.

Backtesting stochastic mortality models by prediction interval-based metrics

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.