Stochastic Mortality Models Research Articles

A Bayesian Approach to Developing a Stochastic Mortality Model for China

Summary Stochastic mortality models have a wide range of applications. They are particularly important for analysing Chinese mortality, which is subject to rapid and uncertain changes. However, owing to data-related problems, stochastic modelling of Chinese mortality has not been given adequate attention. We attempt to use a Bayesian approach to model the evolution of Chinese mortality over time, taking into account all of the problems associated with the data set. We build on the Gaussian state space formulation of the Lee–Carter model, introducing new features to handle the missing data points, to acknowledge the fact that the data are obtained from different sources and to mitigate the erratic behaviour of the parameter estimates that arises from the data limitations. The approach proposed yields stochastic mortality forecasts that are in line with both the trend and the variation of the historical observations. We further use simulated pseudodata sets with resembling limitations to validate the approach. The validation result confirms our approach’s success in dealing with the limitations of the Chinese mortality data.

Beyond the highest life expectancy: construction of proxy upper and lower life expectancy bounds

The strong, persistent linear trend of the highest period life expectancy of females at birth, namely best-performance life expectancy, is an interesting global phenomenon which has already lasted for about 150 years. In this article, we study both the highest and lowest period life expectancies of a group of more developed countries and areas, and exploit their recent trends to construct approximate upper and lower bounds as a supplementary tool for future projections. We also seek two modifications of this proposed approach. First, despite that it has remained largely an empirical observation, we intend to examine the use of extreme value theory to provide a more theoretical framework for both the highest and lowest life expectancies. Second, we construct two hypothetical populations with each age experiencing the lowest or highest mortality rate amongst all the populations considered, and extrapolate their life expectancy trends into the future. The resulting life expectancy bounds perform reasonably well in our backtesting exercise and can potentially complement the usual application of a stochastic mortality model to the data of a single country.

An introduction to gevistic regression mortality models

ABSTRACTMany common stochastic mortality models can be formulated as a generalized linear model (GLM). When these GLMs are used to model one year-death probabilities, , deaths are assumed to be binomially distributed, and the canonical logit link function has been used by default. In this work we present the quantile function of the Generalized Extreme Value distribution as an alternative link function to the standard canonical logit link and show that its theoretical advantages enable a better fit for mortality models in cases when data are highly imbalanced or sparse. We provide an example that shows that this link function also enables superior fits to mortality data at the very highest ages in the case of the Cairns Blake Dowd family of mortality models.

Application of Machine Learning to Mortality Modeling and Forecasting

Estimation of future mortality rates still plays a central role among life insurers in pricing their products and managing longevity risk. In the literature on mortality modeling, a wide number of stochastic models have been proposed, most of them forecasting future mortality rates by extrapolating one or more latent factors. The abundance of proposed models shows that forecasting future mortality from historical trends is non-trivial. Following the idea proposed in Deprez et al. (2017), we use machine learning algorithms, able to catch patterns that are not commonly identifiable, to calibrate a parameter (the machine learning estimator), improving the goodness of fit of standard stochastic mortality models. The machine learning estimator is then forecasted according to the Lee-Carter framework, allowing one to obtain a higher forecasting quality of the standard stochastic models. Out-of sample forecasts are provided to verify the model accuracy.

An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap

The management of National Social Security Systems is being challenged more and more by the rapid ageing of the population, especially in the industrialized countries. In order to chase the Pension System sustainability, several countries in Europe are setting up pension reforms linking the retirement age and/or benefits to life expectancy. In this context, the accurate modelling and projection of mortality rates and life expectancy play a central role and represent issues of great interest in recent literature. Our study refers to the Italian mortality experience and considers an indexing mechanism based on the expected residual life to adjust the retirement age and keep costs at an expected budgeted level, in the spirit of sharing the longevity risk between Social Security Systems and retirees. In order to combine fitting and projections performances of selected stochastic mortality models, a model assembling technique is applied to face uncertainty in model selection, while accounting for uncertainty of estimation as well. The resulting proposal is an averaged model that is suitable to discuss about the gender gap in longevity risk and its alleged narrowing over time.

Reducing Model Risk and Improving Mortality Forecasts for Life Insurance Product Pricing

The pricing of life insurance products depends critically on the ability to model and forecast three core stochastic drivers. Firstly, the ability to accurately forecast expected mortality rates by age group for a given population in order to construct estimates of the life expectancy required for survival linked insurance products. Secondly, the ability to model interest rate dynamics accurately over multi-decade time horizons, and thirdly the ability to model the causal relationship between mortality events and interest rate fluctuations. In this work we tackle all three aspects of these challenging problems faced by actuaries seeking to robustly price life products. We demonstrate with real data for three major populations, U.K., U.S.A. and Australia that we are able to reduce the model risk and associated forecast errors of classical Lee-Carter models in constructing forecasts for mortality and subsequent life expectancy by age and gender. This is achieved by developing new classes of multivariate long-memory models for mortality which we compare to extensions of classical Lee-Carter models. Secondly, we develop standard short rate one factor models for interest rates, in which we incorporate dependence links with our stochastic mortality models. We develop a Bayesian calibration and forecasting framework which is estimated with a Hamiltonian Markov Chain Monte Carlo sampling procedure. We then utilise these frameworks to study the influence of model risk for life products including annuity portfolios and the valuation of a guaranteed annuity option (GAO). We demonstrate that classical Lee-Carter type models can produce less accurate model forecasts than our proposed multivariate long memory models and we quantify the mispricing cost of this model risk.

Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

This paper assesses the hedge effectiveness of an index-based longevity swap and a longevity cap for a life annuity portfolio. Although longevity swaps are a natural instrument for hedging longevity risk, derivatives with non-linear pay-offs, such as longevity caps, provide more effective downside protection. A tractable stochastic mortality model with age dependent drift and volatility is developed and analytical formulae for prices of longevity derivatives are derived. The model is calibrated using Australian mortality data. The hedging of the life annuity portfolio is comprehensively assessed for a range of assumptions for the longevity risk premium, the term to maturity of the hedging instruments, as well as the size of the underlying annuity portfolio. The results compare the risk management benefits and costs of longevity derivatives with linear and nonlinear payoff structures.

Spatial Mortality Modelling With Economic Growth

At present, academic actuarial research involving the mortality modeling of multiple populations mainly focuses on factor-based approaches. This comes with little attention to interpretable models of mortality that take patterns across space into consideration. To address this, we propose a family of models that extend the seminal factor-based stochastic mortality modeling framework of Li and Lee (2005) to include spatial patterns. Specifically, in this paper, we study the relationship between economic growth, as represented by the real gross domestic product (GDP), and mortality of the contiguous United States. The proposed spatial lag of GDP with GDP (SLGG) model was used to produce forecasts of mortality rates and annuity pricing for each of the states of the United States and demonstrated the effects which economic growth has on mortality. A comparison of annuity pricing across space revealed that the SLGG model preserves more regional differences when it comes to pricing compared to the Li and Lee (2005) model. In a larger context, this research provides a blueprint for the inclusion of spatial components and economic growth into mortality modeling. Importantly, it establishes an empirical basis for the development of spatial natural hedging techniques.

Still living with mortality: the longevity risk transfer market after one decade

Abstract This paper updates Living with Mortality published in 2006. It describes how the longevity risk transfer market has developed over the intervening period, and, in particular, how insurance-based solutions – buy-outs, buy-ins and longevity insurance – have triumphed over capital markets solutions that were expected to dominate at the time. Some capital markets solutions – longevity-spread bonds, longevity swaps, q-forwards and tail-risk protection – have come to market, but the volume of business has been disappointingly low. The reason for this is that when market participants compare the index-based solutions of the capital markets with the customised solutions of insurance companies in terms of basis risk, credit risk, regulatory capital, collateral and liquidity, the former perform on balance less favourably despite a lower potential cost. We discuss the importance of stochastic mortality models for forecasting future longevity and examine some applications of these models, e.g. determining the longevity risk premium and estimating regulatory capital relief. The longevity risk transfer market is now beginning to recognise that there is insufficient capacity in the insurance and reinsurance industries to deal fully with demand and new solutions for attracting capital markets investors are now being examined – such as longevity-linked securities and reinsurance sidecars.

Comparison of Stochastic Mortality Model for Wider Age Range

The incorporation of non-linear pattern of early ages has opened new research directions on improving the existing stochastic mortality model structure. Several authors have outlined the importance of encompassing the full age range in dealing with longevity risk exposure by not to ignore the dependence between young and old age. In this study, we consider the two extensions of Cairns, Blake and Dowd model that incorporate the irregularity profile seen at the mortality of lower ages which are Plat and O’Hare and Li. The models’ performances in terms of in-sample fitting and out-sample forecasts were examined and compared. The results indicated that O’Hare and Li model performs better as compared to Plat model

Using the stochastic health state function to forecast healthcare demand and healthcare financing: Evidence from Singapore

AbstractTypically, healthcare financing for an ageing population requires projections on healthcare demand and cost. However, projecting healthcare demand based on projected elderly does not consider changes in population health state over time. This paper proposes a new approach to forecast health variables using a stochastic health state function and the well‐established Lee–Carter stochastic mortality model. With the estimated health state at each age over time, we project the hospitalization rate, healthcare demand, and financing cost for Singapore using historical life tables and hospital admission data. Our findings show that while hospital insurance claims increase owing to an aging population, improving health state could save costs from hospital insurance claims. This has policy implications: more attention should be given to preventive healthcare such as health screening to improve the overall health state of the population.

Forecasting mortality rates: multivariate or univariate models?

It is well known that accurate forecasts of mortality rates are essential to various demographic research topics, such as population projections and the pricing of insurance products such as pensions and annuities. In this study, we argue that including the lagged rates of neighbouring ages cannot further improve mortality forecasting after allowing for autocorrelations. This is because the sample cross-correlation function cannot exhibit meaningful and statistically significant correlations. In other words, rates of neighbouring ages are usually not leading indicators in mortality forecasting. Therefore, multivariate stochastic mortality models like the classic Lee–Carter may not necessarily lead to more accurate forecasts, compared with sophisticated univariate models. Using Australian mortality data, simulation and empirical studies employing the Lee–Carter, Functional Data, Vector Autoregression, Autoregression-Autoregressive Conditional Heteroskedasticity and exponential smoothing (ETS) state space models are performed. Results suggest that ETS models consistently outperform the others in terms of forecasting accuracy. This conclusion holds for both female and male mortality data with different empirical features across various forecasting error measurements. Hence, ETS can be a widely useful tool to model and forecast mortality rates in actuarial practice.

An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing

During the last decades, life expectancy has risen significantly in the most developed countries all over the world. Greece is a case in point; consequently, higher governmental financial responsibilities occur as well as serious concerns are raised owing to population ageing. To address this issue, an efficient forecasting method is required. Therefore, the most important stochastic models were comparatively applied to Greek data for the first time. An analysis of their fitting behaviour by gender was conducted and the corresponding forecasting results were evaluated. In particular, we incorporated the Greek population data into seven stochastic mortality models under a common age-period-cohort framework. The fitting performance of each model was thoroughly evaluated based on information criteria values as well as the likelihood ratio test and their robustness to period changes was investigated. In addition, parameter risk in forecasts was assessed by employing bootstrapping techniques. For completeness, projection results for both genders were also illustrated in pricing insurance-related products.

A stochastic mortality model for annuities to calibrate longevity risk

A stochastic mortality model for annuities to calibrate longevity risk

Longevity Risk Management and the Development of a Value-Based Longevity Index

The design and development of post-retirement income products require the assessment of longevity risk, as well as a basis for hedging these risks. Most indices for longevity risk are age-period based. We develop and assess a cohort-based value index for life insurers and pension funds to manage longevity risk. There are two innovations in the development of this index. Firstly, the underlying variables of most existing longevity indices are based on mortality experience only. The value index is based on the present value of future cash flow obligations, capturing all the risks in retirement income products. We use the index to manage both longevity risk and interest rate risk. Secondly, we capture historical dependencies between ages and cohorts with a cohort-based stochastic mortality model. We achieve this by introducing age-dependent model parameters. With our mortality model, we obtain realistic cohort correlation structures and improve the fitting performance, particularly for very old ages.

StMoMo: An R Package for Stochastic Mortality Modeling

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 goodnessof-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.

Still Living with Mortality: The Longevity Risk Transfer Market After One Decade

This paper updates Living with Mortality published in 2006. It describes how the longevity risk transfer market has developed over the intervening period, and, in particular, how insurance-based solutions – buy-outs, buy-ins and longevity insurance – have triumphed over capital markets solutions that were expected to dominate at the time. Some capital markets solutions – longevity-spread bonds, longevity swaps, q-forwards, and tail-risk protection – have come to market, but the volume of business has been disappointingly low. The reason for this is that when market participants compare the index-based solutions of the capital markets with the customized solutions of insurance companies in terms of basis risk, credit risk, regulatory capital, collateral, and liquidity, the former perform on balance less favourably despite a lower potential cost. We discuss the importance of stochastic mortality models for forecasting future longevity and examine some applications of these models, e.g., determining the longevity risk premium and estimating regulatory capital relief. The longevity risk transfer market is now beginning to recognize that there is insufficient capacity in the insurance and reinsurance industries to deal fully with demand and new solutions for attracting capital markets investors are now being examined – such as longevity-linked securities and reinsurance sidecars.

Retirement spending and biological age

We solve a lifecycle model in which the consumer’s chronological age does not move in lockstep with calendar time. Instead, biological age increases at a stochastic non-linear rate in time like a broken clock that might occasionally move backwards. In other words, biological age could actually decline. Our paper is inspired by the growing body of medical literature that has identified biomarkers which indicate how people age at different rates. This offers better estimates of expected remaining lifetime and future mortality rates. It isn’t farfetched to argue that in the not-too-distant future personal age will be more closely associated with biological vs. calendar age. Thus, after introducing our stochastic mortality model we derive optimal consumption rates in a classic (Yaari, 1965) framework adjusted to our proper clock time. In addition to the normative implications of having access to biological age, our positive objective is to partially explain the cross-sectional heterogeneity in retirement spending rates at any given chronological age. In sum, we argue that neither biological nor chronological age alone is a sufficient statistic for making economic decisions. Rather, both ages are required to behave rationally.

A COMPARATIVE STUDY OF TWO-POPULATION MODELS FOR THE ASSESSMENT OF BASIS RISK IN LONGEVITY HEDGES

AbstractLongevity swaps have been one of the major success stories of pension scheme de-risking in recent years. However, with some few exceptions, all of the transactions to date have been bespoke longevity swaps based upon the mortality experience of a portfolio of named lives. In order for this market to start to meet its true potential, solutions will ultimately be needed that provide protection for all types of members, are cost effective for large and smaller schemes, are tradable, and enable access to the wider capital markets. Index-based solutions have the potential to meet this need; however, concerns remain with these solutions. In particular, the basis risk emerging from the potential mismatch between the underlying forces of mortality for the index reference portfolio and the pension fund/annuity book being hedged is the principal issue that has, to date, prevented many schemes progressing their consideration of index-based solutions. Two-population stochastic mortality models offer an alternative to overcome this obstacle as they allow market participants to compare and project the mortality experience for the reference and target populations and thus assess the amount of demographic basis risk involved in an index-based longevity hedge. In this paper, we systematically assess the suitability of several multi-population stochastic mortality models for assessing basis risks and provide guidelines on how to use these models in practical situations paying particular attention to the data requirements for the appropriate calibration and forecasting of such models.

Stochastic Mortality Modelling: Some Extensions Based on LLvy CARMA Models

Force of mortality is defined using an exponential function of Legendre polynomials, as in Renshaw et al. (1996), plus an extra term which captures mortality shocks. For the extra term Ballotta Haberman (2006) and Ahmadi et al. (2015) consider an Ornstein-Uhlenbeck while we suggest using Levy Continuous Autoregressive Moving Average (CARMA) models. The proposed model encompasses as special cases the existing approaches. We present some extensions in the theory of Levy CARMA models that enter in the construction of the model estimation procedure. Males life tables for USA, Taiwan and Japan for the time period 1998-2013 are used for the presentation of fitting and projection results.