Stochastic Mortality Models Research Articles

Pricing participating longevity-linked life annuities: a Bayesian Model Ensemble approach

Participating longevity-linked life annuities (PLLA) in which benefits are updated periodically based on the observed survival experience of a given underlying population and the performance of the investment portfolio are an alternative insurance product offering consumers individual longevity risk protection and the chance to profit from the upside potential of financial market developments. This paper builds on previous research on the design and pricing of PLLAs by considering a Bayesian Model Ensemble of single population generalised age-period-cohort stochastic mortality models in which individual forecasts are weighted by their posterior model probabilities. For the valuation, we adopt a longevity option decomposition approach with risk-neutral simulation and investigate the sensitivity of results to changes in the asset allocation by considering a more aggressive lifecycle strategy. We calibrate models using Taiwanese (mortality, yield curve and stock market) data from 1980 to 2019. The empirical results provide significant valuation and policy insights for the provision of a cost effective and efficient risk pooling mechanism that addresses the individual uncertainty of death, while providing appropriate retirement income and longevity protection.

Automatic Indexation of the Pension Age to Life Expectancy: When Policy Design Matters

Increasing retirement ages in an automatic or scheduled way with increasing life expectancy at retirement is a popular pension policy response to continuous longevity improvements. The question addressed here is: to what extent is simply adopting this approach likely to fulfill the overall goals of policy? To shed some light on the answer, we examine the policies of four countries that have recently introduced automatic indexation of pension ages to life expectancy–The Netherlands, Denmark, Portugal and Slovakia. To this end, we forecast an alternative period and cohort life expectancy measures using a Bayesian Model Ensemble of heterogeneous stochastic mortality models comprised of parametric models, principal component methods, and smoothing approaches. The approach involves both the selection of the model confidence set and the determination of optimal weights. Model-averaged Bayesian credible prediction intervals are derived accounting for various stochastic process, model, and parameter risks. The results show that: (i) retirement ages are forecasted to increase substantially in the coming decades, particularly if a constant period in retirement is targeted; (ii) retirement age policy outcomes may substantially deviate from the policy goal(s) depending on the design adopted and its implementation; and (iii) the choice of a cohort over period life expectancy measure matters. In addition, the distributional issues arising with the increasing socio-economic gap in life expectancy remain largely unaddressed.

Assessment of longevity risk: credibility approach

To correctly measure the effect of mortality rates on the stability of insurance and pension provider's financial risk, longevity risk should be considered. This paper aims to investigate the future mortality and longevity risk with different age structures for different countries. Lee–Carter mortality model is used on the historical census data to forecast future mortality rates. Turkey, Germany, and Japan are chosen concerning their expected life and population distributions. Then, the longevity risk on a hypothetical portfolio is assessed based on static and dynamic mortality table approaches. To determine the impact of longevity risk, which is retrieved using a stochastic mortality model, a pension insurance product is taken into account. The net single premium for an annuity is quantified under the proposed set up for the selected countries. Additionally, the credibility approach is proposed to establish a reliable estimate for the annuity net single premium.

Mortality data correction in the absence of monthly fertility records

Since the conjecture of Richards (2008), the work by Cairns et al. (2016) and subsequent developments by Boumezoued (2020), Boumezoued et al. (2020) and Boumezoued et al. (2019), it has been acknowledged that observations from censuses have led to major problems of reliability in estimates of general population mortality rates as implemented in practice. These issues led to mis-interpretation of some key mortality characteristics in the past decades, including ”false cohort effects”. To overcome these issues, the exposure estimates for a given country can be corrected by using monthly fertility records. However, in the absence of birth-by-month data, the recent developments are not applicable. Therefore, this paper explores new solutions regarding the construction of mortality tables in this context, based on machine learning techniques. As a main result, it is demonstrated that the new exposure models proposed in this paper allow to provide correction with high quality and to improve the fitting of stochastic mortality models without a cohort component, as it is the case for the existing correction method based on monthly fertility data.

Recent declines in life expectancy: Implication on longevity risk hedging

An article published in the British Medical Journal in 2018 reveals that a number of developed countries have experienced a decline in life expectancy in recent years. Within the classical framework of stochastic mortality modeling, the observed decline in life expectancy may be attributed to noises around the fitted log-linear trends in age-specific death rates. However, the patterns of the mortality heat maps for these countries suggest that it is likely a result of a fading of waves of high mortality improvement, which previously contributed to a linear rise in life expectancy in the developed world. In this paper, we introduce an improved version of the heat wave mortality model, which has the potential to capture the cessation of the waves of high mortality improvement. The proposed model is then used to examine the impact of declines in life expectancy on index-based longevity hedges. It is found that if life expectancy declines, a simple delta hedge still performs reasonably well in the sense that the over-hedging problem is only modest.

Addressing the life expectancy gap in pension policy

Understanding the systematic relationship between period and cohort life expectancy and how the relationship evolves over time are critical issues in formulating the design of retirement income products, evaluating the actuarial balance of pension schemes, and more generally for all analyses where demographic projections are involved. In this study, estimates of the life expectancy gap at all ages are performed using data for 1960–2018 from the Human Mortality Database and projections are generated through 2050 for the 42 national populations, disaggregated by sex. Contrary to previous research that often uses a single deemed to be «best» model to forecast mortality rates, we use a novel adaptive Bayesian Model Ensemble of heterogeneous parametric generalized age-period-cohort stochastic mortality models, principal component methods, and smoothing approaches. The procedure involves both the selection of the model confidence set and the determination of optimal weights. Model-averaged Bayesian credible prediction intervals are derived accounting for both the uncertainty arising from model error and parameter uncertainty. With intergenerational actuarial fairness and neutrality as the guiding principles the study then explores potential policy interventions to address the consequences of the life expectancy gap - spanning over adjustments in the accumulation, benefit determination, and payout stages. Comprehensive numerical results are provided for two policy options: (i) introducing a sustainability factor; and (ii) conditional pension indexation. The results show that: (i) the life expectancy gap is positive and significant for almost all countries and years studied, (ii) it will continue to increase, (iii) the magnitude of the subsidy rates between generations can be sizeable demanding important initial pension benefit reduction and/or a gradual diminution in the annual indexation rate of pensions to correct them.

Assessing mortality inequality in the U.S.: What can be said about the future?

This paper investigates mortality inequality across U.S. states by modeling and forecasting mortality rates via a forecast reconciliation approach. Understanding the heterogeneity in state-level mortality experience is of fundamental importance, as it can assist decision making for policymakers, health authorities, as well as local communities who are seeking to reduce inequalities and disparities in life expectancy. A key challenge of multi-population mortality modeling is high dimensionality, and the resulting complex dependence structures across sub-populations. Moreover, when projecting future mortality rates, it is important to ensure that the state-level forecasts are coherent with the national-level forecasts. We address these issues by first obtaining independent state-level forecasts based on classical stochastic mortality models, and then incorporating the dependence structure in the forecast reconciliation process. Both traditional bottom-up reconciliation and the cutting-edge trace minimization reconciliation methods are considered. Based on the U.S. total mortality data for the period 1969–2017, we project the 10-year-ahead mortality rates at both national-level and state-level up to 2027. We find that the geographical inequality in the longevity levels is likely to continue in the future, and the mortality improvement rates will tend to slow down in the coming decades.

Time-consistent longevity hedging with long-range dependence

Longevity securitization enables insurers to manage longevity or mortality risk in the life market. Recent empirical studies identify long-range dependence (LRD) in mortality rates using life tables, which casts doubt on the adequacy of Markovian models for actuarial pricing and risk management. This paper derives the first time-consistent mean–variance longevity hedging strategy for insurers using a stochastic mortality model with LRD. We adopt the open-loop equilibrium control framework and derive an analytical solution to the hedging strategy. Our numerical examples show the relevance of LRD to longevity hedging and the cost of ignoring it.

Forecasting mortality rates with the penalized exponential smoothing state space model

It is well known that accurate forecasts of mortality rates are essential to various demographic research like population projection, and the pricing of insurance products such as pensions and annuities. Recent studies suggest that mortality rates of multivariate ages are usually not leading indicators in mortality forecasting. Therefore, multivariate stochastic mortality models including the classic Lee–Carter may not necessarily lead to more accurate forecasts, compared with sophisticated univariate counterparties like the exponential smoothing state space (ETS) model. Despite its improved forecasting accuracy, the original ETS model cannot ensure the age-coherence of forecast mortality rates. By introducing an effective penalty scheme, we propose a penalized ETS model to significantly overcome this problem, with discussions on related technical issues including the reduction of parameter dimensionality and the selection of tuning parameter. Empirical results based on mortality rates of the Australian males and females suggest that the proposed model consistently outperforms the Lee–Carter and original ETS models. Robust conclusions are drawn when various forecasting scenarios are considered. Long-term forecasting analyses up to 2050 comparing the three models are further performed. To illustrate its usefulness in practice, an application to price fixed-term annuities with the penalized ETS model is demonstrated.

Two Factor Stochastic Mortality Modeling with Generalized Hyperbolic Distribution

In this paper, we reconsider the two-factor stochastic mortality model introduced by Cairns, Blake and Dowd (2006) (CBD). The error terms in the CBD model are assumed to form a two-dimensional random walk. We first use the Doornik and Hansen (2008) multivariate normality test to show that the underlying normality assumption does not hold for the considered data set. Ainou (2011) proposed independent univariate normal inverse Gaussian Levy processes to model the error terms in the CBD model. We generalize this idea by introducing a possible dependency between the 2-dimensional random variables, using a bivariate Generalized Hyperbolic distribution. We propose four non-Gaussian, fat-tailed distributions: Student's t, normal inverse Gaussian, hyperbolic and generalized hyperbolic distributions. Our empirical analysis shows some preferences for using the new suggested model, based on Akaike's information criterion, the Bayesian information criterion and likelihood ratio test, as our in-sample model selection criteria, as well as mean absolute percentage error for our out-of-sample projection errors.

Two Factor Stochastic Mortality Modeling with Generalized Hyperbolic Distribution

Two Factor Stochastic Mortality Modeling with Generalized Hyperbolic Distribution

Point and Interval Forecasts of Death Rates Using Neural Networks

The 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 architecture for mortality rate forecasting, empirically compare this model as well as other neural network 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, forecasts and global behavior of our model to make it more understandable and, thus, more trustworthy. As neural networks 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 neural network model.

Backcasting Mortality in England and Wales, 1600–1840

There have been significant developments in using extrapolative stochastic models for mortality forecasting (forward projection) in the literature. However, little attention has been devoted to mortality backcasting (backward projection). This article proposes a simple mortality backcasting framework that can be used in practice. Research and analysis of English demography in the 17th and 18th centuries have suffered from a lack of mortality data. We attempt to alleviate this problem by developing a technique that runs backward in time and produces estimates of mortality data before the time at which such data became available. After confirming the time reversibility of the mortality data, we compare the backcasting performance of some commonly used stochastic mortality models for the England and Wales data. The original Lee–Carter model is selected for backcasting purpose of this dataset. Finally, we examine the longevity of British artists between the 17th and the 20th centuries using the backcasted population mortality as benchmarks. The results show that artists living in Britain from 1600 to the mid 1800s had life expectancies similar to those of the general population, with a marked increase in longevity after the Industrial Revolution.

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 the standard and early retirement ages in a discretionary, scheduled, or automatic way, and by 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. In addition, our analyses demonstrate that countries risk falling short of their goals given their use of projection methods that underestimate life expectancy. 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. We show that 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. The empirical strategy employed to project the relevant cohort life expectancy uses a Bayesian Model Ensemble approach to stochastic mortality modelling to generate forecasts of intergenerationally and actuarially fair pension ages for 23 countries and regions from 2000 to 2050. The empirical results 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.

A multi-country comparison of stochastic models of breast cancer mortality with P-splines smoothing approach

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.

A State-Space Estimation of the Lee-Carter Mortality Model and Implications for Annuity Pricing

A common feature of retirement income products is that their payouts depend on the lifetime of policyholders. A typical example is a life annuity policy which promises to provide benefits regularly as long as the retiree is alive. Consequently, insurers have to rely on best estimate life tables, which consist of age-specific mortality rates, in order to price these kind of products properly. Recently there is a growing concern about the accuracy of the estimation of mortality rates since it has been historically observed that life expectancy is often underestimated in the past (so-called longevity risk), thus resulting in longer benefit payments than insurers have originally anticipated. To take into account the stochastic nature of the evolution of mortality rates, Lee and Carter (1992) proposed a stochastic mortality model which primarily aims to forecast age-specific mortality rates more accurately. The original approach to estimating the Lee-Carter model is via a singular value decomposition, which falls into the least squares framework. Researchers then point out that the Lee-Carter model can be treated as a state-space model. As a result several well-established state-space modeling techniques can be applied to not just perform estimation of the model, but to also perform forecasting as well as smoothing. Research in this area is still not yet fully explored in the actuarial literature, however. Existing relevant literature focuses mainly on mortality forecasting or pricing of longevity derivatives, while the full implications and methods of using the state-space representation of the Lee-Carter model in pricing retirement income products is yet to be examined. The main contribution of this article is twofold. First, we provide a rigorous and detailed derivation of the posterior distributions of the parameters and the latent process of the Lee-Carter model via Gibbs sampling. Our assumption for priors is slightly more general than the current literature in this area. Moreover, we suggest a new form of identification constraint not yet utilised in the actuarial literature that proves to be a more convenient approach for estimating the model under the state-space framework. Second, by exploiting the posterior distribution of the latent process and parameters, we examine the pricing range of annuities, taking into account the stochastic nature of the dynamics of the mortality rates. In this way we aim to capture the impact of longevity risk on the pricing of annuities. The outcome of our study demonstrates that an annuity price can be more than4% under-valued when different assumptions are made on determining the survival curve constructed from the distribution of the forecasted mortality rates. Given that a typical annuity portfolio consists of a large number of policies with maturities which span decades, we conclude that the impact of longevity risk on the accurate pricing of annuities is a significant issue to be further researched. In addition, we find that mis-pricing is increasingly more pronounced for older ages as well as for annuity policies having a longer maturity.

Mortality data reliability in an internal model

AbstractIn this paper, we discuss the impact of some mortality data anomalies on an internal model capturing longevity risk in the Solvency 2 framework. In particular, we are concerned with abnormal cohort effects such as those for generations 1919 and 1920, for which the period tables provided by the Human Mortality Database show particularly low and high mortality rates, respectively. To provide corrected tables for the three countries of interest here (France, Italy and West Germany), we use the approach developed by Boumezoued for countries for which the method applies (France and Italy) and provide an extension of the method for West Germany as monthly fertility histories are not sufficient to cover the generations of interest. These mortality tables are crucial inputs to stochastic mortality models forecasting future scenarios, from which the extreme 0.5% longevity improvement can be extracted, allowing for the calculation of the solvency capital requirement. More precisely, to assess the impact of such anomalies in the Solvency II framework, we use a simplified internal model based on three usual stochastic models to project mortality rates in the future combined with a closure table methodology for older ages. Correcting this bias obviously improves the data quality of the mortality inputs, which is of paramount importance today, and slightly decreases the capital requirement. Overall, the longevity risk assessment remains stable, as well as the selection of the stochastic mortality model. As a collateral gain of this data quality improvement, the more regular estimated parameters allow for new insights and a refined assessment regarding longevity risk.

Career breaks, broken pensions? Long-run effects of early and late-career unemployment spells on pension entitlements

AbstractUnemployment periods and other career breaks have long-term scarring effects on future labour market possibilities, permanently affecting workers' retirement income and standard of living as pensioners. Previous literature has focused on the impact of job loss on working careers with little attention to its impact on pension wealth, particularly the extent to which longevity heterogeneity amplifies unemployment scars. This paper investigates the effect of single and multiple unemployment spells on the lifetime pension entitlements of earnings-related contributory pension schemes, considering the timing and duration of breaks, alternative lifecycle labour earnings profiles, scarring and restoration effects on labour market re-entry, the existence of pension credits and pension accruals for periods spent outside the labour market, longevity heterogeneity, and the accumulation and decumulation redistributive features of the pension scheme. Pension entitlements are estimated using a backward-looking simulation approach based on the actual Portuguese public pension system rules and stylized labour market profiles identified in the SHARE Job Episodes Panel data using a sequence analysis. Longevity heterogeneity is modelled using a stochastic mortality model with a frailty model. Our results show that the timing and duration of unemployment periods is critical, that scarring effects amplify pension wealth losses, that minimum pension provisions, pension credits and pension scheme redistributive features can partially mitigate the impact of unemployment periods on future entitlements, and that the presence of positive correlation between lifetime income and longevity career breaks can amplify the asymmetry in the distribution of pension entitlements across income groups.

Estimation of Uncertainty in Mortality Projections Using State-Space Lee-Carter Model

The study develops alternatives of the classical Lee-Carter stochastic mortality model in assessment of uncertainty of mortality rates forecasts. We use the Lee-Carter model expressed as linear Gaussian state-space model or state-space model with Markovian regime-switching to derive coherent estimates of parameters and to introduce additional flexibility required to capture change in trend and non-Gaussian volatility of mortality improvements. For model-fitting, we use a Bayesian Gibbs sampler. We illustrate the application of the models by deriving the confidence intervals of mortality projections using Lithuanian and Swedish data. The results show that state-space model with Markovian regime-switching adequately captures the effect of pandemic, which is present in the Swedish data. However, it is less suitable to model less sharp but more prolonged fluctuations of mortality trends in Lithuania.

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 rolling” 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 the 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 hedging effectiveness. We benchmark our strategy against the delta” hedging strategy as well as its robustness to q-forwards’ maturity, reference age, 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.