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.     

On the Modeling and Forecasting of Socioeconomic Mortality Differentials: An Application to Deprivation and Mortality in England

In any country, mortality rates and indices such as life expectancy usually differ across subpopulations, for example, defined by gender, geographic area, or socioeconomic variables (e.g., occupation, level of education, or income). These differentials, and in particular those related to socioeconomic circumstances, pose important challenges for the design of public policies for tackling social inequalities, as well as for the design of pension systems and the management of longevity risk in pension funds and annuity portfolios. We discuss the suitability for the modeling and forecasting of socioeconomic differences in mortality of several multiple population extensions of the Lee-Carter model, including a newly introduced relative model based on the modeling of the mortality in socioeconomic subpopulations alongside the mortality of a reference population. Using England mortality data for socioeconomic subpopulations defined using a deprivation index, we show that this new relative model exhibits the best results in terms of goodness of fit and ex post forecasting performance. We then use this model to derive projections of deprivation specific mortality rates and life expectancies at pensioner ages and analyze the impact of socioeconomic differences in mortality on the valuation of annuities.     

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.     

Mortality Change and Forecasting

Abstract

Prospects of longer life are viewed as a positive change for individuals and as a substantial social achievement but have led to concern over their implications for public spending on old-age support. This paper makes a critical assessment of knowledge about mortality change. It is oriented toward the problem of forecasting the course of mortality change and the potential of existing work to contribute to the development of useful forecasts in Canada, Mexico, and the U.S.

We first examine broad patterns in the historical decline in death rates in the three countries, the effect of these on trends in life expectancy, and the epidemiological transition. Next we review theories of the age pattern and evolution of mortality, including graduations, evolutionary theory, reliability models, dynamic models, and relational models.

The analysis and forecasting of mortality change have been shaped largely by some key historical lessons, which we summarize next. We emphasize issues that have been or are likely to be significant in mortality analysis, especially the questions of the age pattern and time trend in mortality at old ages; we distinguish patterns and facts that are established from those that remain uncertain. Next, we consider mortality differentials in characteristics such as sex, marital status, education, and socioeconomic variables; we summarize their key features and also point to the substantial gaps in our understanding of their determinants.

Finally, we review methods of forecasting, including the scenario method used by the U.S. Social Security Administration and the time series method of Lee and Carter. We set out some important recommendations for forecasters: forecasting assumptions should be made more formal and explicit; there should be retrospective evaluations of forecast performance; and greater attention should be paid to the assessment and consequences of forecast uncertainty.     

Explaining Mortality Dynamics

Abstract

Using data for six OECD countries over the period 1950–2006, this paper studies the impact of macroeconomic fluctuations and cause of death trends on mortality dynamics in the Lee-Carter mortality forecasting model. The key results of this study are the following: (1) Periods can be identified in which the Lee-Carter mortality index k_t correlates significantly with macroeconomic fluctuations. (2) A few causes of death such as diseases of the circulatory system, influenza and pneumonia, and diabetes mellitus account for a large fraction of the variations in the Lee-Carter mortality index k_t . (3) Most cause-specific mortality rates show pronounced trends over the last few decades. These trends change the composition of deaths and alter how total mortality reacts to external factors such as macroeconomic fluctuations.     

The evolution of death rates and life expectancy in Denmark

From 1835 to date Denmark has experienced an increase in life expectancy at birth of about 40 years for both sexes. Over the course of the last 170 years, life expectancy at birth has increased from 40 to 80 years for women and from 36 to 76 years for men, and it continues to rise. Using a new methodology, we show that about half of the total historic increase can be attributed to the sharp decline in infant and young age death rates up to 1950. However, life expectancy gains from 1950 to date can be primarily attributed to improvements in the age-specific death rates for the age group from 50 to 80, although there is also a noticeable contribution from the further decline in infant mortality over this period. With age-specific death rates up to age 60 now at a very low absolute level, substantial future life expectancy improvements must necessarily arise from improvements in age-specific death rates for ages 60 and above. Using the developed methodology, we quantify the impact of further reductions in age-specific mortality. Despite being one of countries with the highest life expectancy at the beginning of the 20th century, and despite the spectacular historic increase in life expectancy since then, Denmark is, in fact, lagging behind compared to many other countries, notably the other Nordic countries. The main reason is an alarming excess mortality for cause-specific death rates related to ischaemic heart diseases and, in particular, a number of cancer diseases. Age-specific death rates continue to improve in most countries, and a likely scenario is that in the future Denmark will experience improvement rates at the international level or perhaps even higher as a result of a catch-up effect.     

Life expectancy--a commentary on this life table variable

In 1992, I wrote an article on a method of modifying the Decennial US Life Table to accommodate any pattern of excess mortality expressed in terms of excess death rate (EDR), for the specific purpose of calculating the reduced life expectancy, e. I believe this was the first article published in the Journal of Insurance Medicine (JIM) that dealt specifically with life expectancy as an index of survival and risk appraisal, never used in the classification of extra mortality risk in applicants for life insurance. In this commentary, I discuss the 1989-91 US Decennial Life Table in detail. I link the subject matter of the 1992 article with several more recent articles that also focus on the utility of life expectancy in underwriting structured settlement annuities and preparing reports on life expectancy for an attorney in a tort case. A few references are given for further reading on life table methodology and its use in the most accurate estimate of life expectancy, given the inherent limitations of the life table and the limited duration of follow-up studies.     

Modeling disease elimination

The effect of the elimination of mortality from heart disease and cancer was modelled mathematically to allow for the effect of other competing causes of death. The model allows for potential dependence between heart disease or cancer and other causes of death by using cupola functions, which analyse the individual risk itself and the dependence structure between causes of death by using correlation coefficients. As the strength of these risk associations is unknown, the study investigated both full positive and negative dependence and compared this with no dependence. Depending upon the degree and type of correlation assumed, positive or negative, the life expectancy at birth is increased by between 3 months and 6.5 years if cancer mortality was eliminated, and between 5 months and 7.5 years in the case of heart disease. In addition, estimates of these effects on life insurance premia can be made with the greatest reduction for women with the elimination of cancer mortality. These figures provide a range of improvements in life expectancy and the consequent effect on life insurance risk premium rates which elimination of either of these important diseases would produce.     

Mortality,Health, and Marriage: A Study Based on Taiwan's Population Data

Life expectancy has been increasing significantly since the start of the 20th century, and mortality improvement trends are likely to continue in the 21st century. Stochastic mortality models are used frequently to predict the expansion in life expectancy. In addition to gender, age, period, and cohort are the three main risk factors considered in constructing mortality models. Other than these factors, it is also believed that marital status is related to health and longevity, and many studies have found that married persons have a lower mortality rate than the unmarried. In this study, we have used Taiwan's marital data for the whole population (married, unmarried, divorced/widowed) to evaluate if the marital status can be a preferred criteria. Furthermore, we also want to know whether the preferred criteria will be valid in the future. We chose two popular mortality models, the Lee-Carter and age-period-cohort, to model the mortality improvements for various marital statuses. Because of a linear dependence in the parameters of the age-period-cohort model, we used a computer simulation to choose the appropriate estimation method. Based on Taiwan's marital data, we found that married persons have significantly lower mortality rates than the single, and if converting the difference into a life insurance policy, the discount amount is even larger than that for smokers/nonsmokers.     

Minimum Death Rates and Maximum Life Expectancy: The Role of Concordant Ages

Only five populations have achieved maximum life expectancy (or best practice population) more than occasionally since 1900. The aim of this article is to understand how maximum life expectancy is achieved in the context of mortality transition. We explore this aim using the concepts of potential life expectancy, based on minimum rates at each age among all high longevity populations, and concordant ages. Concordant ages are defined as ages at which the minimum death rate occurs in the population with the maximum life expectancy. The results show the extent to which maximum life expectancy could increase through the realization of demonstrably achievable minimum rates. Concordant ages are concentrated at increasingly older ages over time, but they have produced more than half of the change in maximum life expectancy in almost all periods since 1900. This finding is attributed to their quantity and position whereby concordant ages are concentrated at the ages that have the greatest impact on mortality decline in a particular period. Based on mortality forecasts, we expect that concordant ages will continue to lead increases in female maximum life expectancy, but that they will play a weaker role in male maximum life expectancy.     

How to prepare a life expectancy report for an attorney in a tort case

The purpose of this methodology article is to describe a suitable format for a legally acceptable report on the life expectancy of the principal in a tort case that is being advocated or defended by an attorney. Life insurance medical directors and underwriters are clearly skilled and experienced in mortality risk classification for life insurance. However, the judicial system is accustomed to measuring excess mortality only in terms of reduced life expectancy. The analyst preparing the report must convert the excess mortality into a figure for reduced life expectancy and compare this with the life expectancy of persons matched by age, sex and race in the latest Decennial US Life Tables. This process is different from the life insurance underwriting process. A life table projected to age 109 must be constructed as an essential part of the report, and the entire process must be presented clearly and convincingly. There are good reasons why the excess death rate (EDR) should be used as the index of excess mortality in constructing the life table, in preference to the mortality ratio (MR), which is used most of the time in life insurance risk classification. All of these considerations are discussed in this article, which is based on a sample of 40 cases handled by the author, a retired life insurance medical director.     

How long might the younger-old live: A predictive model

Life expectancy amongst older people in industrialised countries has been improving over an extended period and still continues to do so. This has ramifications for providers of services to this population, thus necessitating a level of forward planning. Predictive models of remaining life expectancy for older age groups can assist long-term planning processes. This paper presents an extrapolative approach to forecasting remaining life expectancy. Based on logistic modelling of historic mortality and survivorship for the “younger-old” male population of England and Wales over the period 1970-2005, a parsimonious two-parameter model is derived. This model provides a close correspondence to published period life table data. Trends in these parameters are then fitted and extrapolated to enable projections of life expectancy up to 40 years into the future. Alternative assumptions are used to determine a range of future life expectancy trajectories for a 65-year-old male. Occupational pension scheme provision is identified as an area of particular concern in the context of increasing longevity. As an illustration, the life expectancy trajectories are combined with differing discount rate assumptions to generate a number of alternative pension liability scenarios for the extrapolation period.     

Der Einfluss des demographischen Wandels auf die Ausgaben der Krankenversicherung

The analysis presented in this document investigates the question of whether the increase in life expectancy causes financial stress for health insurance systems or not. In particular, the authors focused on the financial impact of the ?costs of dying“ and how much these costs contribute to the total health costs. The article analyses an in-patient and an out-patient tariff of a large private health insurance company in Germany. It is based on health care costs of people who died in 1999 and of those who continued to live. The percentage of the costs of dying is often overestimated. However, the costs of those who continued to live increased overproportionately. In particular, this was true for the insured people over 80 years.The claim that the increase of life expectancy only postpones the high costs in the future and has no impact on the financial conditions of health insurance is doubtful. Older people live longer and have more opportunity to take medication and receive therapy for a longer period. Therefore we argue that longer life expectancy and other factors like progress in medical technology pose a severe threat on the financial stability of health insurance.     

Estimation of future mortality rates and life expectancy in chronic medical conditions

Estimates of old-age mortality are necessary for the construction of life tables and computation of life expectancy, and are essential in the growing area of life insurance for the elderly. Two common assumptions are that either the excess death rate (EDR) or the relative risk (RR) stays constant with increasing age. It is known, however, that for most medical conditions the former underestimates the risk and the latter overestimates it. A third popular method is that of rating up: a subject is said to be "rated up k years" if his future mortality rates are assumed to be those of a person in the general population who is k years older. It is shown here that this method generally leads to gross overestimates of old-age mortality. We consider two less-commonly used models, log-linear declining relative risk (LDR) and constant proportional life expectancy (PLE), and compare them to the methods of constant EDR, constant RR and rating up. Although slightly more complicated to employ than the other methods, both LDR and PLE generally give better estimates of mortality and life expectancy. When mortality rates for chronic conditions are known within a certain age range, and estimates outside of the range are required, the LDR and PLE methods may be preferable to the more familiar methods of constant EDR, constant RR, or rating up.     

Mortality regimes and longevity risk in a life annuity portfolio

This paper explores the presence of changes of trends or jumps in French mortality from 1947 to 2007, and assesses their implications on the longevity risk management of a life annuity portfolio. We accomplish this by extending the Poisson log-bilinear regression developed by Brouhns et al. (2002) with a regime-switching model. Estimation results show that French mortality is characterized by two distinct regimes. One refers to a strong uncertainty state, which corresponds to the longevity conditions observed during the decade following World War II. The second regime is related to the low volatility of longevity improvements observed during the last 30 years. We use these results to analyze the impact of mortality regimes on the longevity risk management of a life annuity portfolio. Simulation results suggest that the changes of trends in the mortality process have some implications for longevity risk management.     

Life Expectancy in 2040: What Do Clinical Experts Expect?

We use expert clinical and public health opinion to estimate likely changes in the prevention and treatment of important disease conditions and how they will affect future life expectancy. Focus groups were held including clinical and public health faculty with expertise in the six leading causes of death in the United States. Mortality rates and life tables for 2040 were derived by sex and age. Life expectancy at age 20 and 65 was compared to figures published by the Social Security Administration and to estimates from the Lee-Carter method. There was agreement among all three approaches that life expectancy at age 20 will increase by approximately one year per decade for females and males between now and 2040. According to the clinical experts, 70% of the improvement in life expectancy will occur in cardiovascular disease and cancer, while in the last 30 years most of the improvement has occurred in cardiovascular disease. Expert opinion suggests that most of the increase in life expectancy will be attributable to the already achieved reduction in smoking rates, especially for women.     

The Demand for Life Insurance in OECD Countries

This article examines the determinants of life insurance consumption in OECD countries. Consistent with previous results, we find a significant positive income elasticity of life insurance demand. Demand also increases with the number of dependents and level of education, and decreases with life expectancy and social security expenditure. The country's level of financial development and its insurance market's degree of competition appear to stimulate life insurance sales, whereas high inflation and real interest rates tend to decrease consumption. Overall, life insurance demand is better explained when the product market and socioeconomic factors are jointly considered. In addition, the use of GMM estimates helps reconcile our findings with previous puzzling results based on inconsistent OLS estimates given heteroscedasticity problems in the data.     

Detecting Common Longevity Trends by a Multiple Population Approach

Recently the interest in the development of country and longevity risk models has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare, and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups or the dependence structure across time in a single population setting—a sort of intradependence structure. On the other hand, the dependence across multiple populations, which we describe as interdependence, can be explored for capturing common long-run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee-Carter (LC) framework with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intragroup error correlation structure are preserved. We introduce a model which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current article, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee-Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.     

“A Direct Approach to the Discounted Penalty Function”, Hansjörg Albrecher,Hans U. Gerber,and Hailiang Yang,Volume 14, No. 4, 2010

Abstract

Mortality improvements, especially of the elderly, have been a common phenomenon since the end of World War II. The longevity risk becomes a major concern in many countries because of underestimating the scale and speed of prolonged life. In this study we explore the increasing life expectancy by examining the basic properties of survival curves. Specifically, we check if there are signs of mortality compression (i.e., rectangularization of the survival curve) and evaluate what it means to designing annuity products. Based on the raw mortality rates, we propose an approach to verify if there is mortality compression. We then apply the proposed method to the mortality rates of Japan, Sweden, and the United States, using the Human Mortality Database. Unlike previous results using the graduated mortality rates, we found no obvious signs that mortality improvements are slowing down. This indicates that human longevity is likely to increase, and longevity risk should be seriously considered in pricing annuity products.     

Temporal Evolution of Mortality Indicators

Abstract

In Spain, as in other developed countries, significant changes in mortality patterns have occurred during the 20th and 21st centuries. One reflection of these changes is life expectancy, which has improved in this period, although the robustness of this indicator prevents these changes from being of the same order as those for the probability of death. If, moreover, we bear in mind that life expectancy offers no information as to whether this improvement is the same for different age groups, it is important and necessary to turn to other mortality indicators whose past and future evolution in Spain we are going to study. These indicators are applied to Spanish mortality data for the period 1981–2008, for the age range 0–99. To study its future evolution, the mortality ratios have to be projected using an adequate methodology, namely, the Lee-Carter model. Confidence intervals for these predictions can be calculated using the methodology that Lee and Carter apply in their original article for expected lifetime confidence intervals, but they take into account only the error in the prediction of the mortality index obtained from the ARIMA model adjusted to its temporal series, excluding other sources of error such as that introduced by estimations of the other parameters in the model. That is why bootstrap procedures are preferred, permitting the combination of all sources of uncertainty.