Beyond the Gompertz law: exploring the late-life mortality deceleration phenomenon

Using a new distribution capable of exhibiting all the possible modes of accelerating and decelerating mortality, we conduct a systematic investigation of late-life mortality in humans. We check the insensitivity of the distribution to age cutoffs in the data relative to the logistic mortality model and propose a method to forecast evolution in the characteristic deceleration ages of the distribution. A number of data sets have been explored, with a particular emphasis on those originating from Scandinavia. Although those from Australia, Canada, and the USA are compatible with Gompertzian mortality, those from the other countries examined are not. We find in particular that the onset of mortality deceleration is being progressively delayed in Western societies but that there is evidence of mortality plateauing at earlier ages.     

Forecasting mortality

Abstract

The work of actuaries is concerned with estimating the future on the basis of past experience. The calculation of a premium to be charged for a given risk implies a forecast of the future but so far as mortality is concerned we have generally been content to examine past experience and assume that the results will be repeated. Judged as forecasts our estimates have sometimes been wide of the mark and owing to an almost continuous improvement in mortality actuaries have been assuming heavier rates of mortality than have been experienced. We may defend the use of past experience by saying that it is on the safe side when we are calculating premiums and we may argue that it is the best practical method; but an alternative is to make a more accurate forecast and then allow in our calculations a margin for chance deviations, emergencies, etc. Moreover the assumption that the past will be repeated has not been uniformly safe; it has led to bad results in annuity business and may prove unfortunate in social insurance, pension funds, and even sickness insurance. For some of these purposes we should either work on an estimate of future rates of mortality or, which comes to much the same thing, take a sufficient margin to cover the error involved in our assumptions.     

Stochastic mortality models: an infinite-dimensional approach

Demographic projections of future mortality rates involve a high level of uncertainty and require stochastic mortality models. The current paper investigates forward mortality models driven by a (possibly infinite-dimensional) Wiener process and a compensated Poisson random measure. A major innovation of the paper is the introduction of a family of processes called forward mortality improvements which provide a flexible tool for a simple construction of stochastic forward mortality models. In practice, the notion of mortality improvements is a convenient device for the quantification of changes in mortality rates over time, and enables, for example, the detection of cohort effects. We show that the forward mortality rates satisfy Heath–Jarrow–Morton-type consistency conditions which translate to conditions on the forward mortality improvements. While the consistency conditions for the forward mortality rates are analogous to the classical conditions in the context of bond markets, the conditions for the forward mortality improvements possess a different structure. Forward mortality models include a cohort parameter besides the time horizon, and these two dimensions are coupled in the dynamics of consistent models of forward mortality improvements. In order to obtain a unified framework, we transform the systems of Itô processes which describe the forward mortality rates and improvements. In contrast to term structure models, the corresponding stochastic partial differential equations (SPDEs) describe the random dynamics of two-dimensional surfaces rather than curves.     

The maximum entropy mortality model: forecasting mortality using statistical moments

ABSTRACT

The age-at-death distribution is a representation of the mortality experience in a population. Although it proves to be highly informative, it is often neglected when it comes to the practice of past or future mortality assessment. We propose an innovative method to mortality modeling and forecasting by making use of the location and shape measures of a density function, i.e. statistical moments. Time series methods for extrapolating a limited number of moments are used and then the reconstruction of the future age-at-death distribution is performed. The predictive power of the method seems to be net superior when compared to the results obtained using classical approaches to extrapolating age-specific-death rates, and the accuracy of the point forecast (MASE) is improved on average by 33% respective to the state-of-the-art, the Lee–Carter model. The method is tested using data from the Human Mortality Database and implemented in a publicly available R package.     

A logistic two-population mortality projection model for modelling mortality at advanced ages for both sexes

In the current literature, numerous mortality projection models have been proposed and tested, but in general they have been designed for and applied to mainly ages below 90. As medical advances are being shifted to older ages over time and there is a rapid growth in the number of centenarians, there is a need to expand the modelling to older ages. We propose a logistic two-population mortality projection model for the death rates at ages 80 to 100+ for both sexes. We apply this model and its extensions to high quality old-age mortality data of Belgium, Sweden, Switzerland, and the UK and produce decent model performance in both mortality fitting and forecasting. The model structure also provides a reasonable way to close off the life table, which is supported by both theoretical arguments and empirical evidence.     

The Lee-Carter quantile mortality model

The Lee-Carter (LC) stochastic mortality model has been widely used for making future projections of mortality rates. In the framework of the LC model, the response function is non-linear in parameters. Here, we adapt this LC framework to compute conditional quantiles. The LC quantile model can be defined as quantile non-linear regression conditioned to age and the calendar year. Two strategies for estimating coefficients based on interior-point methods are described. We show that the LC quantile model provides additional information to that furnished by the traditional LC conditional mean. An application to Spanish mortality data is reported.     

Modeling cause-of-death mortality using hierarchical Archimedean copula

Studying changes in cause-specific (or competing risks) mortality rates may provide significant insights for the insurance business as well as the pension systems, as they provide more information than the aggregate mortality data. However, the forecasting of cause-specific mortality rates requires new tools to capture the dependence among the competing causes. This paper introduces a class of hierarchical Archimedean copula (HAC) models for cause-specific mortality data. The approach extends the standard Archimedean copula models by allowing for asymmetric dependence among competing risks, while preserving closed-form expressions for mortality forecasts. Moreover, the HAC model allows for a convenient analysis of the impact of hypothetical reduction, or elimination, of mortality of one or more causes on the life expectancy. Using US cohort mortality data, we analyze the historical mortality patterns of different causes of death, provide an explanation for the ‘failure’ of the War on Cancer, and evaluate the impact on life expectancy of hypothetical scenarios where cancer mortality is reduced or eliminated. We find that accounting for longevity improvement across cohorts can alter the results found in existing studies that are focused on one single cohort.     

Rethinking age-period-cohort mortality trend models

Longevity risk arising from uncertain mortality improvement is one of the major risks facing annuity providers and pension funds. In this article, we show how applying trend models from non-life claims reserving to age-period-cohort mortality trends provides new insight in estimating mortality improvement and quantifying its uncertainty. Age, period and cohort trends are modelled with distinct effects for each age, calendar year and birth year in a generalised linear models framework. The effects are distinct in the sense that they are not conjoined with age coefficients, borrowing from regression terminology, we denote them as main effects. Mortality models in this framework for age-period, age-cohort and age-period-cohort effects are assessed using national population mortality data from Norway and Australia to show the relative significance of cohort effects as compared to period effects. Results are compared with the traditional Lee–Carter model. The bilinear period effect in the Lee–Carter model is shown to resemble a main cohort effect in these trend models. However, the approach avoids the limitations of the Lee–Carter model when forecasting with the age-cohort trend model.     

Continuous-time multi-cohort mortality modelling with affine processes

Continuous-time mortality models, based on affine processes, provide many advantages over discrete-time models, especially for financial applications, where such processes are commonly used for interest rate and credit risks. This paper presents a multi-cohort mortality model for age-cohort mortality rates with common factors across cohorts as well as cohort-specific factors. The mortality model is based on well-developed and used techniques from interest rate theory and has many applications including the valuation of longevity-linked products. The model has many appealing features. It is a multi-cohort model that describes the whole mortality surface, it captures cohort effects, it allows for observed imperfect correlation between different cohorts, it is shown to fit historical data at pension-related ages very well, it has closed-form expressions for survival curves and we show that it outperforms a number of other commonly used discrete-time mortality models in forecasting future survival curves.     

Modeling total cholesterol as predictor of mortality: the low-cholesterol paradox

Elevated total cholesterol is well-established as a risk factor for coronary artery disease and cardiovascular mortality. However, less attention is paid to the association between low cholesterol levels and mortality--the low cholesterol paradox. In this paper, restricted cubic splines (RCS) and complex survey methodology are used to show the low-cholesterol paradox is present in the laboratory, examination, and mortality follow-up data from the Third National Health and Nutrition Examination Survey (NHANES III). A series of Cox proportional hazard models, demonstrate that RCS are necessary to incorporate desired covariates while avoiding the use of categorical variables. Valid concerns regarding the accuracy of such predictive models are discussed. The one certain conclusion is that low cholesterol levels are markers for excess mortality, just as are high levels. Restricted cubic splines provide the necessary flexibility to demonstrate the U-shaped relationship between cholesterol and mortality without resorting to binning results. Cox PH models perform well at identifying associations between risk factors and outcomes of interest such as mortality. However, the predictions from such a model may not be as accurate as common statistics suggest and predictive models should be used with caution.     

Annuitization and aggregate mortality risk

It is well established that annuities can fully diversify idiosyncratic mortality risks. However, survival rates at the cohort level are changing, raising the question what is the scope of annuities in the presence of aggregate mortality risk? In an overlapping generations setting, we show that risk free annuities exist, but offer a return below the (fair) certainty equivalent return, and agents do not fully annuitize their savings. Higher aggregate mortality risk increases savings and thus the mean level of the capital stock. This lowers the mean rate of return on capital, the survival premium on annuities and the share of individual savings in annuities.     

Obstructive sleep apnea and cardiovascular disease and mortality: the argument for causality

There is solid evidence linking obstructive sleep apnea (OSA) to cardiovascular mortality. Although it has yet to be scientifically proven that OSA causes cardiovascular disease, many investigators consider it an independent cardiovascular risk factor. Its impact on the cardiovascular mortality risk of a given applicant varies depending upon the severity of the condition, compliance with treatment, and the applicant's specific cardiovascular milieu. This review is aimed at making mortality risk assessment more accurate by describing what is known of the physiologic mechanisms by which OSA may influence cardiovascular mortality and providing an appreciation for the magnitude of this risk. In doing so, an argument supporting OSA as a cause for cardiovascular disease and mortality emerges.     

On systematic mortality risk and risk-minimization with survivor swaps

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time, this new class of financial contract is interesting from an investor's point of view, since it increases the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically, this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk, we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.     

On the mortality in Danish burial societies

Abstract

In Denmark existed in the years 1900—1940 a union of Burial Societies named »Genforsikringen af Begravelseskasser i Danmark» (the Reinsurance of B. S.'s in Denmark), including in 1940 about 1225 societies with 1)40,000 members. In the following pages we shall, for abbreviation, call it The Reinsurance. This union comprised, according to its byelaws, only B. S.'s connected with sickness societies belonging to the unions for such societies existing in the different parts of the kingdom and almost all receiving grants from the State. As such sickness societies are only intended for the less well-off part of the population, of which the majority are workers, we get rather good information on the mortality in this part of the people by researches on the mortality in the membership of the aforesaid burial societies.     

Infant mortality: an insured population perspective

Many insurers offer life coverage to individuals during the first year of life. The policies tend to have small face values, but frequently contain premium waiver or additional purchase options. General population mortality is significantly higher at this age relative to older children and even middle-aged adults. This article presents the mortality experience of an insured cohort in which death occurred under 1 year of age. In summary, the insured population's mortality rate was significantly lower and the leading causes of death were different than the general population.     

Incorporating the Bühlmann credibility into mortality models to improve forecasting performances

In this paper, we incorporate the Bühlmann credibility into three mortality models (the Lee–Carter model, the Cairns–Blake–Dowd model, and a linear relational model) to improve their forecasting performances, as measured by the MAPE (mean absolute percentage error), using mortality data for the UK. The results show that the MAPE reduction ratios for the three mortality models with the Bühlmann credibility are all significant. More importantly, the MAPEs under the three mortality models with the Bühlmann credibility are very close to each other for each age and forecast year. Thus, by incorporating the Bühlmann credibility we are able to converge the forecasting MAPEs resulting from the three different mortality models to a lower and more consistent level. Moreover, we provide a credibility interpretation with an individual time trend for age x and a group time trend for all ages. Finally, we apply the forecasted mortality rates both with and without the Bühlmann credibility to the net single premiums of life insurance products, and compare the corresponding MAPEs.     

Methods for evaluating mortality risk

An important aspect of decisions related to the long-range planning of large-scale systems, eg world energy supplies, is their possible environmental side-effects. These decisions, which potentially affect the public's health and safety, increasingly require a formal consideration of mortality risk. Several methods have been proposed for the evaluation of public programmes which probabilistically alter human mortality. This article reviews the relevant literature and the practical applications of these proposed methods. Particular emphasis is placed upon the explicit identification of the social objectives implied by their use.     

Pricing implications of trends in population mortality and underwriting effectiveness

Pricing actuaries try to anticipate insured lives mortality rates for decades into the future by considering historic relationships between population and insured lives mortality and trends in population mortality. The degree to which underwriting might decrease insured lives mortality relative to population mortality is of particular importance. A comparison of trends in population and insured mortality is presented to illustrate historic relationships. Two theories for future life expectancy trends are: 1) no foreseeable limit to life expectancy, and 2) life expectancy limited by biological forces. Factors that may increase or decrease the future effectiveness of underwriting are reviewed.     

A Bayesian non-parametric model for small population mortality

This paper proposes a Bayesian non-parametric mortality model for a small population, when a benchmark mortality table is also available and serves as part of the prior information. In particular, we extend the Poisson-gamma model of Hardy and Panjer to incorporate correlated and age-specific mortality coefficients. These coefficients, which measure the difference in mortality levels between the small and the benchmark population, follow an age-indexed autoregressive gamma process, and can be stochastically extrapolated to ages where the small population has no historical exposure. Our model substantially improves the computation efficiency of existing two-population Bayesian mortality models by allowing for closed form posterior mean and variance of the future number of deaths, and an efficient sampling algorithm for the entire posterior distribution. We illustrate the proposed model with a life insurance portfolio from a French insurance company.