An Extreme Value Analysis Of Advanced Age Mortality Data

Abstract

Extreme value theory describes the behavior of random variables at extremely high or low levels. The application of extreme value theory to statistics allows us to fit models to data from the upper tail of a distribution. This paper presents a statistical analysis of advanced age mortality data, using extreme value models to quantify the upper tail of the distribution of human life spans.

Our analysis focuses on mortality data from two sources. Statistics Canada publishes the annual number of deaths in Canada, broken down by angender and age. We use the deaths data from 1949 to 1997 in our analysis. The Japanese Ministry of Health, Labor, and Welfare also publishes detailed annual mortality data, including the 10 oldest reported ages at death in each year. We analyze the Japanese data over the period from 1980 to 2000.

Using the r-largest and peaks-over-threshold approaches to extreme value modeling, we fit generalized extreme value and generalized Pareto distributions to the life span data. Changes in distribution by birth cohort or over time are modeled through the use of covariates. We then evaluate the appropriateness of the fitted models and discuss reasons for their shortcomings. Finally, we use our findings to address the existence of a finite upper bound on the life span distribution and the behavior of the force of mortality at advanced ages.     

Statistical Independence and Fractional Age Assumptions

Abstract

This paper considers in some detail the issue of statistical independence of the curtate future lifetime and the fractional part of the future lifetime of a general status.

Statistical independence is often employed in actuarial contexts, primarily because it leads to simple relationships between quantities of interest and statistical information that is of a discrete nature, such as a life table. The uniform distribution of deaths (UDD) assumption is the most commonly used because of its simplicity and intuitive appeal, but it can be somewhat restrictive. For example, all deaths or withdrawals may be assumed to be at a particular point in the year such as the middle; assumptions of this type are often made in a multiple decrement context. This paper attempts to unify these assumptions and extend their applicability in an actuarial context.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (Fl) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended easily to the general Fl case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the Fl case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the Fl assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the Fl assumption.     

The estimation of mortality and other decremental probabilities

Abstract

The standard actuarial methods of estimating the age-specific one-year probabilities of death in a given community were developed—for the most part, many years ago-with large bodies of observations in mind. Although the familiar “exposed to risk” procedure is known to provide unbiased estimates only when a rather dubious assumption is made about the progression of the instantaneous death-rate (the force of mortality) over the year of age (Cantelli, 1914) it is still the most widely used method of estimation. This is partly because the age-to-age increment in human mortality is relatively small—so that assumptions about its mathematical form are unimportant—and partly because suggested methods of estimation based on more “realistic” assumptions are usually laborious to apply to thousands of observations.     

Life Insurance Mathematics with Random Life Tables

Abstract

When the insurer sells life annuities, projected life tables incorporating a forecast of future longevity must be used for pricing and reserving. To fix the ideas, the framework of Lee and Carter is adopted in this paper. The Lee-Carter model for mortality forecasting assumes that the death rate at age x in calendar year t is of the form exp(α_x + (β_xK_t), where the time-varying parameter K_t reflects the general level of mortality and follows an ARIMA model. The future lifetimes are all influenced by the same time index K_t in this framework. Because the future path of this index is unknown and modeled as a stochastic process, the policyholders' lifetimes become dependent on each other. Consequently the risk does not disappear as the size of the portfolio increases: there always remains some systematic risk that cannot be diversified, whatever the number of policies. This paper aims to investigate some aspects of actuarial mathematics in the context of random life tables. First, the type of dependence existing between the insured life lengths is carefully examined. The way positive dependence influences the need for economic capital is assessed compared to mutual independence, as well as the effect of the timing of deaths through Bayesian credibility mechanisms. Then the distribution of the present value of payments under a closed group of life annuity policies is studied. Failing to account for the positive dependence between insured lifetimes is a dangerous strategy, even if the randomness in the future survival probabilities is incorporated in the actuarial computations. Numerical illustrations are performed on the basis of Belgian mortality statistics. The impact on the distribution of the present value of the additional variability that results from the Lee-Carter model is compared with the traditional method of mortality projection. Also, the impact of ignoring the dependence hat arises from the model is quantified.     

A note on the connection between factorial series distribution and zero-truncated power series distribution

Summary

The main purpose of this note is to call attention to the close relationship between the problem of estimating the parameter of a zero-truncated power series distribution and the problem of estimating the parameter of an associated factorial series distribution. We also extend a known recurrence property for the Poisson case to the binomial and negative binomial cases.     

Analytical studies in stop-loss reinsurance

Abstract

Introduction.

Consider a unit of risk, say the whole portfolio of an office, or a comprehensive contract of a branch of casualty insurance, which can give rise to a variety of total amounts of claims during a chosen period, say one year. The total claims of the years i =- 1, 2, ... will be denoted by x ₁. They follow some frequency distribution and we assume that during the years considered they are independent from year to year and subject to the same parent distribution. This means, implicitly, that the volume of business and the value of money have remained unaltered and this assumption will be made, since the adjustments otherwise needed are technically trivial and we are not dealing here with the commercial aspect (dif. ficult though it may be of solution) arising out of changes in monetary value. The frequency distribution mentioned can then be regarded as given by a sample from a population whose probability distribution is given by p (x), say, so that      

Presentation and Derivation of a Five-Parameter Survival Function Intended to Model Mortality in Modern Female Populations

A single analytical expression for the probability of survival from birth to age x that would hold good for all ages in the entire human life span has been sought for centuries. With an eight-parameter function, Heligman and Pollard achieved this goal, for integer values of x , in 1980. The present paper introduces a five-parameter survival function intended to model human mortality in modern female populations. The introduced function is not only defined for integers. It is defined for all ages.     

Converting Clinical Literature to an Insured Population: A Comparison of Models Using Nhanes

Abstract

The use of clinical literature to set risk classification standards for life insurance underwriting stems from the need to set the most accurate standards using the best available information. A necessary hurdle in this process is converting any excess mortality observed in a clinical study to the appropriate rating for use in underwriting. A widely accepted model in the insurance industry, the Excess Death Rate model, treats the excess as additive to the conditional probability of death for an insurance company’s unimpaired class.

In this paper we test the validity of that model versus other common predictive models of excess mortality in an insured population. Applying these models to National Health and Nutrition Examination Survey (NHANES) data, we derive estimates for excess mortality from three commonly seen underwriting impairments in what could be considered a clinical population. These estimates are added to an estimate of an insurance company’s unimpaired mortality class and then used to predict deaths in an “insurable” subset of that clinical population.

The Excess Death Rate model performed the best of all models, having the smallest cumulative difference of actual to predicted deaths. The use of publicly available data, such as that in NHANES, could help bridge the gap between clinical literature and its application in insurance underwriting if insurable cohorts can be reliably identified from these generally healthy, ambulatory groups.     

A Hermite-spline model of post-retirement mortality

ABSTRACT

We present a model for post-retirement mortality where differentials automatically reduce with increasing age, but without the fitted mortality rates for subgroups crossing over. Selection effects are catered for, as are age-modulated time trends and seasonal variation in mortality. Central to the model are Hermite splines, which permit parsimonious modelling of complex risk factors in even modest-sized portfolios. The model is therefore suitable for the stand-alone analysis of experience data for reinsurance, bulk annuities and longevity swaps. We also illustrate the contrast between the statistical significance of a risk factor and its financial significance and discuss reasons why one might include risk factors like season that are not directly financially significant.     

“Long-Term Yield Rates for Actuarial Valuations”, Jacques F. Carriere,July, 1999

Abstract

This paper explores the relative significance of aging as a determinant of financial cost of health care beyond age fifty, with particular attention to the effect for ages 65 and over. The paper presents concepts of aging factors and aging curves, which define relative values between ages for utilization or cost of health care services. General conclusions are drawn from Medicare data that utilization and cost differ by age, but that aging factors vary across services and may be less significant at the very old ages. The author then turns to the question of measuring the significance of an aging curve assumption and what accuracy is lost in the simpler alternative of a single value across an age range. The practical effects of relative value aging curves are examined through hypothetical examples of increasing complexity in a retiree health valuation. A method to measure the impact is put forth. Three important variables are discussed in some detail. A survey to ascertain aging curve findings and preferences of health actuaries is introduced and discussed, with one representative curve presented. This curve is then measured to understand its impact vis-à-vis other curves.

It may be important to note this is not a study deriving a recommended aging curve. Rather, it is an exploration of the significance of an assumption that has not received much public actuarial scrutiny. A conclusion places the paper-s findings in a context of a dynamic health care economy in an aging society.     

Robust portfolio estimation under skew-normal return processes

In this paper, we study issues related to the optimal portfolio estimators and the local asymptotic normality (LAN) of the return process under the assumption that the return process has an infinite moving average (MA) (∞) representation with skew-normal innovations. The paper consists of two parts. In the first part, we discuss the influence of the skewness parameter δ of the skew-normal distribution on the optimal portfolio estimators. Based on the asymptotic distribution of the portfolio estimator ? for a non-Gaussian dependent return process, we evaluate the influence of δ on the asymptotic variance V(δ) of ?. We also investigate the robustness of the estimators of a standard optimal portfolio via numerical computations. In the second part of the paper, we assume that the MA coefficients and the mean vector of the return process depend on a lower-dimensional set of parameters. Based on this assumption, we discuss the LAN property of the return's distribution when the innovations follow a skew-normal law. The influence of δ on the central sequence of LAN is evaluated both theoretically and numerically.     

Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method

Abstract

In almost all stochastic claims reserving models one assumes that accident years are independent. In practice this assumption is violated most of the time. Typical examples are claims inflation and accounting year effects that influence all accident years simultaneously. We study a Bayesian chain ladder model that allows for accounting (calendar) year effects modeling. A case study of a general liability dataset shows that such accounting year effects contribute substantially to the prediction uncertainty and therefore need a careful treatment within a risk management and solvency framework.     

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.     

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.     

Approaches and Experiences in Projecting Mortality Patterns for the Oldest-Old

Abstract

In 1998 the United Nations Population Division extended the age format of its estimates and projections of population dynamics for all countries and areas of the world from 80 years and above to 100 years and above. The paper is based on experiences made during the implementation of relevant mortality projection methodologies and their application in two rounds of global population projections.

The paper first briefly addresses the need for the explicit inclusion of very old population segments into the regular UN estimates and projections. It is argued that since population aging is an important issue for both developed and developing countries, the need for more information regarding the elderly, and the oldest-old in particular, is significant.

The paper then documents the methods that have been evaluated and implemented, namely, the relational mortality standard proposed by Himes, Preston, and Condran, the Coale-Kisker extrapolation method for extending empirical age patterns of mortality to very high ages, and the Carter-Lee projection method for projecting model patterns of mortality to very high levels of life expectancy at birth. The methods are critically reviewed, and possible improvements to the methods are discussed.

The paper concludes with a discussion of different views regarding the future evolution of mortality at older ages, their regional variability, and the necessity to improve the coverage and quality of data collected in this area.     

Mortality Measurement at Advanced Ages

Abstract

Accurate estimates of mortality at advanced ages are essential to improving forecasts of mortality and the population size of the oldest old age group. However, estimation of hazard rates at extremely old ages poses serious challenges to researchers: (1) The observed mortality deceleration may be at least partially an artifact of mixing different birth cohorts with different mortality (heterogeneity effect); (2) standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high at old ages and (3) ages of very old people may be exaggerated. One way of obtaining estimates of mortality at extreme ages is to pool together international records of persons surviving to extreme ages with subsequent efforts of strict age validation. This approach helps researchers to resolve the third of the above-mentioned problems but does not resolve the first two problems because of inevitable data heterogeneity when data for people belonging to different birth cohorts and countries are pooled together. In this paper we propose an alternative approach, which gives an opportunity to resolve the first two problems by compiling data for more homogeneous single-year birth cohorts with hazard rates measured at narrow (monthly) age intervals. Possible ways of resolving the third problem of hazard rate estimation are elaborated. This approach is based on data from the Social Security Administration Death Master File (DMF). Some birth cohorts covered by DMF could be studied by the method of extinct generations. Availability of month of birth and month of death information provides a unique opportunity to obtain hazard rate estimates for every month of age. Study of several single-year extinct birth cohorts shows that mortality trajectory at advanced ages follows the Gompertz law up to the ages 102–105 years without a noticeable deceleration. Earlier reports of mortality deceleration (deviation of mortality from the Gompertz law) at ages below 100 appear to be artifacts of mixing together several birth cohorts with different mortality levels and using cross-sectional instead of cohort data. Age exaggeration and crude assumptions applied to mortality estimates at advanced ages may also contribute to mortality underestimation at very advanced ages.     

A dynamic bivariate common shock model with cumulative effect and its actuarial application

Abstract

Standard actuarial theory of multiple life insurance traditionally postulates independence for the remaining lifetimes mainly due to computational convenience rather than realism. In this paper, we propose a general common shock model for modelling dependent coupled lives and apply it to a life insurance model. In the proposed shock model, we consider not only simultaneous deaths of the coupled members due to a single shock (e.g. a critical accident), but also cumulative effect in the mortality rate when they survive shocks. Under the model, we derive a bivariate lifetime distribution and its marginal distributions in closed forms. We study the bivariate ageing property, dependence structure and the dependence orderings of the lifetime distribution. Based on it, we investigate the influence of dependence on the pricings of insurance policies involving multiple lives which are subject to common shocks. Furthermore, we discuss relevant useful stochastic bounds.     

Extreme value theory versus traditional GARCH approaches applied to financial data: a comparative evaluation

Although stock prices fluctuate, the variations are relatively small and are frequently assumed to be normally distributed on a large time scale. But sometimes these fluctuations can become determinant, especially when unforeseen large drops in asset prices are observed that could result in huge losses or even in market crashes. The evidence shows that these events happen far more often than would be expected under the generalised assumption of normally distributed financial returns. Thus it is crucial to model distribution tails properly so as to be able to predict the frequency and magnitude of extreme stock price returns. In this paper we follow the approach suggested by McNeil and Frey in 2000 and combine GARCH-type models with the extreme value theory to estimate the tails of three financial index returns – S&P 500, FTSE 100 and NIKKEI 225 – representing three important financial areas in the world. Our results indicate that EVT-based conditional quantile estimates are more accurate than those from conventional GARCH models assuming normal or Student's t distribution innovations when doing not only in-sample but also out-of-sample estimation. Moreover, these results are robust to alternative GARCH model specifications. The findings of this paper should be useful to investors in general, since their goal is to be able to forecast unforeseen price movements and take advantage of them by positioning themselves in the market according to these predictions.     

Some remarks on the effect of selection upon mortality with regard to different causes of death

Abstract

In a previous number of this Journal ¹ Zum Studium der Sterblichkeit minderwertiger Leben. Skandinavisk Aktuarietidskrift 1921, p. 31. , I have given an account of an investigation into the increased mortality among certain groups of sub-standard risks based only on statistics of deaths. The material, collected for that investigation, can also be used as basis for an attempt of investigating the influence of Selection on the mortality among stan,dard risks.