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.     

The Metabolic Syndrome and All-Cause Mortality in an Insured Lives Population

Abstract

Metabolic syndrome and its association with mortality have not been studied in insured lives populations. The Swiss Re Study evaluated metabolic syndrome prevalence and associated mortality from all causes and circulatory disease in a cohort of 35,470 predominantly healthy individuals, aged 18–83 years, who were issued life insurance policies between 1986 and 1997. Metabolic syndrome was defined using the National Cholesterol Education Program (NCEP) Expert Panel Adult Treatment Panel (ATP) III guidelines. The NCEP obesity criteria were modified with a prediction equation using body mass index, gender, and age substituted for waist circumference. Adjustments also were made for nonfasting triglyceride and blood glucose values. Risk ratios for policyholders identified with metabolic syndrome were 1.16 (P = .156) for mortality from all causes and 1.45 (P = .080) for mortality from circulatory disease compared with individuals without the syndrome. Risk was proportional to the number of components, or score, of the metabolic syndrome present. Risk ratios for metabolic syndrome score were 1.14 (P < .001) for mortality from all causes and 1.38 (P < .001) for mortality from circulatory disease compared with individuals without metabolic syndrome factors. In both all-cause and circulatory death models, relative risk was highest for the blood pressure risk factor. Based on a modified NCEP definition, increased mortality risk is associated with metabolic syndrome in an insured lives cohort and has life insurance mortality pricing implications.     

Infantile mortality from an actuarial point of view

Abstract

1. The difficulties by which the actuary is confronted in dealing with infantile mortality resemble in several ways the difficulties experienced at extreme old age. In both cases the variation in the mortality is so rapid that the usual formulas of numerical approximation may break down. In both cases the observations of the life insurance companies are scarce, and hypothesis must therefore be resorted to. In both cases there is a question of convergence, if the force of mortality is assumed to tend to infinity, as the age tends to zero. In both cases it is, of course, impossible to tell with certainty whether the force of mortality does actually tend to infinity.     

Select mortality and other durational effects modelled by partially observed Markov Chains

Abstract

In a recent paper Norberg explained select mortality tables for insured lives by a simple Markov model where the lives are classified as active/disabled and insured/not insured, and where no return is possible to previously visited states. The present paper extends the set-up and its results to more complex state spaces and patterns of transition, the key tool being the Kolmogorov backward differential equations.     

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.     

“Economic Capital Allocation Derived from Risk Measures”, Eddy Van den Borre,January 2003

Abstract

The sustained reduction in mortality rates and its systematic underestimation has been attracting the significant interest of researchers in recent times because of its potential impact on population size and structure, social security systems, and (from an actuarial perspective) the life insurance and pensions industry worldwide. Despite the number of papers published in recent years, a comprehensive review has not yet been developed.

This paper attempts to be the starting point for that review, highlighting the importance of recently published research—most of the references cited span the last 10 years—and covering the main methodologies that have been applied to the projection of mortality rates in the United Kingdom and the United States. A comparative review of techniques used in official population projections, actuarial applications, and the most influential scientific approaches is provided. In the course of the review an attempt is made to identify common themes and similarities in methods and results.

In both official projections and actuarial applications there is some evidence of systematic overestimation of mortality rates. Models developed by academic researchers seem to reveal a trade-off between the plausibility of the projected age pattern and the ease of measuring the uncertainty involved. The Lee-Carter model is one approach that appears to solve this apparent dilemma.

There is a broad consensus across the resulting projections: (1) an approximately log-linear relationship between mortality rates and time, (2) decreasing improvements according to age, and (3) an increasing trend in the relative rate of mortality change over age. In addition, evidence suggests that excessive reliance on expert opinion—present to some extent in all methods—has led to systematic underestimation of mortality improvements.     

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.     

Large sample validity of the binomial distribution for lives with unequal mortality rates

Abstract

Let us consider a group of n lives which are observed during some time or age interval. Suppose that the following conditions are satisfied: 1. The probability of death within the interval considered has the same value q for each person of the group.

2. These lives represent statistically independent observations (with respect to mortality).

     

Mortality Regimes and Pricing

Abstract

Mortality dynamics are characterized by changes in mortality regimes. This paper describes a Markov regime-switching model that incorporates mortality state switches into mortality dynamics. Using the 1901-2005 U.S. population mortality data, we illustrate that regime-switching models can perform better than well-known models in the literature. Furthermore, we extend the 1992 Lee-Carter model in such a way that the time-series common risk factor to all cohorts has distinct mortality regimes with different means and volatilities. Finally, we show how to price mortality securities with this model.     

On the construction of tables for the calculation of certain survivorship benefits

Abstract

1. Benefits depending on two lives often involve the necessity for calculating tables with two arguments. There are, however, cases where this can be avoided, the chief example being the joint-life annuity when the mortality table is graduated by Makeham's formula. The question therefore arises whether there are other cases where a suitable graduation of the mortality table may render similar services.     

Mortality Among the Elderly in Sweden 1988–1997

By replacing the exponential growth in a Makeham function with a straight line at very high ages, graduated mortality rates gives an acceptable adherence to observed data.     

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.     

Market Value of Liabilities Mortality Risk

Abstract

Market values of the invested assets are frequently published. For most insurance liabilities, there are no published market values and, therefore, these have to be constructed. This construction can be based on a best estimate and a price for the risks in the liabilities. This paper presents a model explaining how the best estimate and the price of mortality risk can be constructed. Several methods to describe the risks are already known. The purpose of this paper is to describe a method to determine the mortality risk in a practical way.     

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.     

Mortality-Indexed Annuities Managing Longevity Risk Via Product Design

Abstract

Longevity risk has become a major challenge for governments, individuals, and annuity providers in most countries. In its aggregate form, the systematic risk of changes to general mortality patterns, it has the potential for causing large cumulative losses for insurers. Since obvious risk management tools, such as (re)insurance or hedging, are less suited for managing an annuity provider’s exposure to this risk, we propose a type of life annuity with benefits contingent on actual mortality experience.

Similar adaptations to conventional product design exist with investment-linked annuities, and a role model for long-term contracts contingent on actual cost experience can be found in German private health insurance. By effectively sharing systematic longevity risk with policyholders, insurers may avoid cumulative losses.

Policyholders also gain in comparison with a comparable conventional annuity product: Using a Monte Carlo simulation, we identify a significant upside potential for policyholders while downside risk is limited.     

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.     

Seasonal mortality for fractional ages in short term life insurance

Abstract

Fernández-Durán, and Gregorio-Domínguez, Seasonal Mortality for Fractional Ages in Life Insurance. Scandinavian Actuarial Journal. A uniform distribution of deaths between integral ages is a widely used assumption for estimating future-lifetimes; however, this assumption does not necessarily reflect the true distribution of deaths throughout the year. We propose the use of a seasonal mortality assumption for estimating the distribution of future-lifetimes between integral ages: this assumption accounts for the number of deaths that occurs in given months of the year, including the excess mortality that is observed in winter months. The impact of this seasonal mortality assumption on short-term life insurance premium calculations is then examined by applying the proposed assumption to Mexican mortality data.     

A Model for Analyzing the Impact of Selective Lapsation on Mortality

Abstract

This paper presents a model for examining the effect of various relationships between mortality rates and lapse rates on the mortality experience of a cohort of insured lives. The approach is individual rather than the aggregate traditionally used in analyzing selective lapsation. The model assumes that insured lives are healthy at policy issue, but later may move to an impaired state from which the lapse rate is zero. Associated with each insured is an unobservable “risk level” random variable, which reflects the heterogeneity of the insured group. Individual mortality and lapse rates are functions of the risk level. A numerical illustration provides some interesting results obtained by using this model.     

The mortality among Norwegian assured lives 1925–1935

Abstract

The most important work the Statistical Bureau of the Norwegian Life Assurance Companies has to do, is the periodical investigations of the mortality among the different groups of the companies policyholders.     

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.